Plasma-Material Interaction in Controlled Fusion (Springer Series on Atomic, Optical, and Plasma Physics)

Springer Series on

atomic, optical, and plasma physics

39

Springer Series on

atomic, optical, and plasma physics The Springer Series on Atomic, Optical, and Plasma Physics covers in a comprehensive manner theory and experiment in the entire f ield of atoms and molecules and their interaction with electromagnetic radiation. Books in the series provide a rich source of new ideas and techniques with wide applications in f ields such as chemistry, materials science, astrophysics, surface science, plasma technology, advanced optics, aeronomy, and engineering. Laser physics is a particular connecting theme that has provided much of the continuing impetus for new developments in the f ield. The purpose of the series is to cover the gap between standard undergraduate textbooks and the research literature with emphasis on the fundamental ideas, methods, techniques, and results in the f ield.

27 Quantum Squeezing By P.D. Drumond and Z. Ficek 28 Atom, Molecule, and Cluster Beams I Basic Theory, Production and Detection of Thermal Energy Beams By H. Pauly 29 Polarization, Alignment and Orientation in Atomic Collisions By N. Andersen and K. Bartschat 30 Physics of Solid-State Laser Physics By R.C. Powell (Published in the former Series on Atomic, Molecular, and Optical Physics) 31 Plasma Kinetics in Atmospheric Gases By M. Capitelli, C.M. Ferreira, B.F. Gordiets, A.I. Osipov 32 Atom, Molecule, and Cluster Beams II Cluster Beams, Fast and Slow Beams, Accessory Equipment and Applications By H. Pauly 33 Atom Optics By P. Meystre 34 Laser Physics at Relativistic Intensities By A.V. Borovsky, A.L. Galkin, O.B. Shiryaev, T. Auguste 35 Many-Particle Quantum Dynamics in Atomic and Molecular Fragmentation Editors: J. Ullrich and V.P. Shevelko 36 Atom Tunneling Phenomena in Physics, Chemistry and Biology Editor: T. Miyazaki 37 Charged Particle Traps Physics and Techniques of Charged Particle Field Confinement By V.N. Gheorghe, F.G. Major, G. Werth 38 Plasma Physics and Controlled Nuclear Fusion By K. Miyamoto 39 Plasma-Material Interaction in Controlled Fusion By D. Naujoks Vols. 1–26 of the former Springer Series on Atoms and Plasmas are listed at the end of the book

D. Naujoks

Plasma-Material Interaction in Controlled Fusion With 54 Figures and 11 Tables

123

Dr. Dirk Naujoks Max-Planck-Institut f¨ur Plasmaphysik Teilinstitut Greifswald 17491 Greifswald, Germany E-mail: [email protected]

ISSN 1615-5653 ISBN-10 3-540-32148-9 Springer Berlin Heidelberg New York ISBN-13 978-3-540-32148-4 Springer Berlin Heidelberg New York Library of Congress Control Number: 2006927044 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media. springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. A X macro package Typesetting by the authors and SPi using a Springer LT E Cover concept by eStudio Calmar Steinen Cover design: design & production GmbH, Heidelberg

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to Natalie

Preface

Nuclear fusion has the potential to provide a major part of mankind’s energy needs for many millennia. On the way to controlled thermonuclear fusion on our planet, the principal goals—from the physical point of view—are firstly to obtain a sufficiently stable plasma, secondly, to heat this plasma to ignition temperature, and finally, to avoid excessive interaction of the hot plasma with the solid wall of the containing vessel. With respect to the foreseen use of a fusion reactor as an energy-producing device, an ideal plasma confinement is unattainable nor is it desired. The generated energy together with the helium particles (fusion “ash”) must be removed from the central region and conducted to the energy exchanging facilities (blanket), as well as to the gas exhausting and purifying systems—at the rate they are produced. The so-called first wall, the border between the hot plasma with sun-like parameters (and beyond) and the “cool earth”, should be able to withstand the high energy and particle fluxes with little or no maintenance. During the last few decades a large number of dedicated experimental results as well as theoretical and simulation studies have been performed— thanks to the effort of scientists from many countries participating in this truly international project of controlled fusion. Several aspects of plasma–surface interaction have been reviewed in various publications [1–5] as well as in proceedings of conferences such as the series of Plasma Surface Interaction and Fusion Reactor Materials conferences. A comprehensive “Data Compendium” related to atomic processes taking place in plasma–surface interactions and material questions is given in special supplements issued by the journal Nuclear Fusion [6–9]. I tried to provide an in-depth look at the multi-faceted aspects of plasma– surface interaction in controlled fusion, to give a comprehensive analysis of the main processes and the main parameters ruling them, together with an assessment of the most critical questions and open points that demand further investigation. I hope this can assist the reader by performing their own estimations and assessments of relevant physical processes and problems. For further

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Preface

studies, references to selected papers are given. A comprehensive review of the enormous experimental work done in this field is out of the scope of this book, but can be found in the references given above. A more detailed quantitative analysis can be acquired by applying simulation techniques, which are presented shortly, used together with special data compilations. Since the involved processes are identical, the book might also be of interest in the fast-paced field of surface modification by means of plasma technology. Whether thin layers are deposited on materials in order to improve the surface characteristics or whether plasma ions are implanted into the depth using biased targets, the underlying physics is the same as in fusion experiments. I would like to thank Prof. V. N. Afanas’ev for conducting my initial steps into scientific work in the field of particle interaction with solids, who has encouraged me to combine experimental, theoretical, and simulation studies wherever it seems possible. He suffered, but dealt gracefully with my poor Russian during my stay in Moscow. I am much obliged to Dr. R. Behrisch, who has introduced me to the world of plasma–surface interaction. He allowed me to benefit from his wide experience and pushed me to make things clear and simple without unessential elaborating—not always with success I am afraid. I have also learned from him to fight for each discharge against the persistent fear of the operators of how a rather small surface probe would harm the device by exposing it into the plasma. I am thankful to Prof. G. Fussmann, who surprised me with analytical descriptions of effects, which I thought were studied only by computer simulation, and who showed me, in detail, that the complexity of particle– surface interaction is quite negligible compared to the situation in another topic—in plasma physics. The suggestions and corrections made by both of them are very appreciated and helped to improve the manuscript significantly. Thanks to all colleagues I worked with in the inspiring and challenging atmosphere of the international fusion community. I am grateful to Dr. Ascheron for his encouragement and support to publish this book with Springer, and in particular, to Ms. Blanck for her excellent technical assistance. Many thanks to Ms. Dewitz for most of the drawings and illustrations.

Greifswald, May 2006

Dirk Naujoks

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Part I Fusion as Energy Source 2

Energy Problem and Related Safety Aspects . . . . . . . . . . . . . .

5

3

Fusion Fuel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.1 Fusion Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Ignition and Burn Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4

Fusion Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Inertial Plasma Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Magnetic Plasma Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Stellarator Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Tokamak Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Design of the First Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Limiter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Divertor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 20 20 22 23 25 26

Part II The Plasma-Material Interface 5

The Plasma State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Ionization Degree and Coupling Constant . . . . . . . . . . . . . . . . . . . 5.2 Debye Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Plasma Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Collisions in Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Transport Processes in Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Transport by Binary Collisions . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Neoclassical Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Anomalous Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 32 34 35 36 41 42 43 44

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5.6 The Vlasov Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.7 The Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6

Particle Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.1 Binary Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.1.1 Scattering Angle α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.1.2 Scattering in the Coulomb Field, U (r) = C/r . . . . . . . . . 57 6.1.3 Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.1.4 Interaction Potential U (r) . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.1.5 Binary Collision: General Case . . . . . . . . . . . . . . . . . . . . . . 62 6.2 Particle Transport in Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.2.1 Definitions and Main Parameters . . . . . . . . . . . . . . . . . . . . 66 6.2.2 Elastic Energy Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.2.3 Inelastic Energy Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.3 Material Modification by Ion Beams . . . . . . . . . . . . . . . . . . . . . . . 74 6.4 Retention and Tritium Inventory Control . . . . . . . . . . . . . . . . . . . 77 6.5 Impurity Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.5.1 Physical Sputtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.5.2 Chemical Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.5.3 Radiation-Enhanced Sublimation . . . . . . . . . . . . . . . . . . . . 87 6.5.4 Thermal Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.5.5 Blistering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.6 Charge Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.7 Diffusion-Controlled Sputtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6.8 Backscattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.8.1 One-Collision Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.8.2 The Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.8.3 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.9 Electron Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.9.1 Secondary Electron Emission (SEE) . . . . . . . . . . . . . . . . . 99 6.9.2 Thermionic Electron Emission . . . . . . . . . . . . . . . . . . . . . . 100 6.9.3 Electron Emission by the Application of an Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.10 Modeling of Particle–Solid Interaction . . . . . . . . . . . . . . . . . . . . . . 102 6.10.1 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.10.2 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7

Electrical Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.1 Electron Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.2 Ion Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.3 Bohm Criterion with the “=” Sign . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.4 Space Charge Limited Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 7.5 Effect of Magnetic Field Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 118

Contents

XI

7.6 Modeling of the Electric Sheath . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.6.1 Principles of PIC Simulations . . . . . . . . . . . . . . . . . . . . . . . 120 7.6.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.6.3 Choice of Time Step and Spatial Resolution . . . . . . . . . . 124 8

Power Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.1 Heat Flux Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.2 Change of Surface Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 8.2.1 Heat Conduction in a Half-Infinite Medium . . . . . . . . . . . 130 8.2.2 Point-like Heat Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8.2.3 Heat Conduction and Diffusion . . . . . . . . . . . . . . . . . . . . . . 131 8.3 Power Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 8.4 Thermal Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

9

Impurity Problems in Fusion Experiments . . . . . . . . . . . . . . . . . 135 9.1 Impurity Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 9.1.1 Line Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 9.1.2 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 9.1.3 Cyclotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 9.1.4 Radiation Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 9.1.5 Benefits of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 9.2 Erosion Phenomena in Fusion Experiments . . . . . . . . . . . . . . . . . 141 9.2.1 Plasma Disruption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 9.2.2 Edge Localized Modes (ELMs) . . . . . . . . . . . . . . . . . . . . . . 143 9.2.3 Runaway Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 9.2.4 Erosion by Energetic Alpha Particles . . . . . . . . . . . . . . . . 145 9.2.5 Hot Spots or Carbon “Blooming” . . . . . . . . . . . . . . . . . . . 146 9.2.6 Flake and Dust Production . . . . . . . . . . . . . . . . . . . . . . . . . 147 9.2.7 Erosion by Charge-Exchange Neutrals . . . . . . . . . . . . . . . . 147 9.2.8 Erosion by Arcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 9.2.9 Non-Linear Erosion due to Impurities . . . . . . . . . . . . . . . . 149 9.3 Impurity Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 9.3.1 Spatial Distributions of Neutrals . . . . . . . . . . . . . . . . . . . . 156 9.3.2 Atomic Processes in Impure Plasmas . . . . . . . . . . . . . . . . . 161 9.3.3 Prompt Redeposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 9.3.4 SOL Screening Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 9.3.5 Accumulation of High-Z Impurities . . . . . . . . . . . . . . . . . . 167 9.3.6 Transport Barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 9.3.7 Sawteeth as Plasma Cleaner . . . . . . . . . . . . . . . . . . . . . . . . 168 9.3.8 Deposition of Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 9.3.9 Modeling of Erosion and Redeposition . . . . . . . . . . . . . . . 172 9.4 Critical Impurity Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

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Part III Operation Limits and Criteria 10 The Problem of Plasma Density Control . . . . . . . . . . . . . . . . . . . 181 10.1 Long-Term Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 10.2 Wall Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 11 Plasma Operation Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 12 Material Operation Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 12.1 Erosion Flux into the Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 12.2 Impurity Density in the Plasma Core . . . . . . . . . . . . . . . . . . . . . . 197 12.3 Impurity Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 12.4 Lifetime of Wall Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 12.4.1 Simple Geometrical Model of Redeposition . . . . . . . . . . . 203 12.4.2 Net Erosion at Divertor Plates . . . . . . . . . . . . . . . . . . . . . . 205 12.4.3 Net Erosion at Wall Plates . . . . . . . . . . . . . . . . . . . . . . . . . 208 12.5 Neutron Irradiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 13 Choice of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 13.1 Candidates of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 13.1.1 Discussion of Plasma-Facing Materials . . . . . . . . . . . . . . . 218 13.1.2 Construction Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 13.2 Alternative Concepts and Innovative Ideas . . . . . . . . . . . . . . . . . . 220 13.3 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 14 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 A.1 Some Important Relations and Parameters . . . . . . . . . . . . . . . . . 231 A.2 Simple Particle Mover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 A.3 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 A.4 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 A.5 Fundamental Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . 247 A.6 Physical Properties of Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

1 Introduction

In plasma–material interactions the fourth state of matter, the plasma, meets the first, the solid state, generally leaving out the states in between, i.e., the gaseous and the liquid state. Serious problems arise. The solid is subjected to power load and to bombardment of energetic particles from the plasma. Due to the contact, the plasma loses energy and particles. At the same time, it is cooled by low-energy plasma particles in the boundary layer and by wall atoms entering the plasma. Such situations are found in the cosmos at the surface of objects with no atmosphere, such as on the moon, which is loaded by the solar wind, or at space vehicles and dust particles in the sun system (for example in the rings of Saturn) and interstellar space. The ignition of electric arcs and discharges on the surfaces of satellites can cause partial or full failure of their sensitive electronics. Before the first astronauts set their feet on the moon surface there had been some fears that the moon could be covered by thick layers of dust produced via erosion processes due to particle impact. A special investigation was launched to estimate the particle fluxes and to get an idea of the corresponding sputtering yields. As we all know, Neil Armstrong had “both feet on the ground”. The dust layer on the moon is rather thin—as was estimated. On our planet, plasma is produced in electric gas discharges in a large number of applications and in experiments dedicated to controlled fusion. The vessel walls in such experimental devices are bombarded with energetic ions, electrons, and neutrals causing the release of wall material atoms that penetrate as impurities into the plasma. Due to this erosion, the lifetime of wall components is limited. Radiation damage is inflicted on the wall material by high-energy neutrons created in the fusion reactions in an ignited fusion plasma. Fusion represents a very difficult challenge from the viewpoint of developing the required materials in order to meet the demands for economic, safe and environmentally acceptable fusion power reactors. Besides long pulse operation, high power and particle fluxes, short transient events such as disruptions, ELMs, hot spots, and arcing could damage the wall considerably.

2

1 Introduction

The description of the strong, entirely mutual, interaction of a plasma with a solid will require significant improvements in the detailed understanding of the different processes taking place in the plasma–solid transition zone: • The power may be reflected or absorbed. It diffuses into the bulk and will be removed at the rear side by cooling. The surface temperatures may rise to high values leading to melting and sublimation. • The bombarding particles may be backscattered into the plasma. They may also be implanted into the wall material and diffuse further into the bulk or towards the surface, where they can be thermally desorbed back into the plasma. The particles may be trapped in the solid permanently, and, therefore, would be lost in the recycling process. • Particle bombardment causes a release of surface atoms from the solid, i.e., sputtering. This results in an introduction of wall atoms into the plasma, where they are converted to impurity ions, resulting in fuel dilution and strong energy losses due to radiation. • Electrons hitting the surface of the solid are mostly implanted, but a fraction may also be backscattered, or secondary electrons may be released. • The loss of electrons from the plasma causes the build-up of an electric potential between the plasma and the solid, the Langmuir sheath potential. In this electrical field the impinging ions increase their velocity, and subsequently, their ability to sputter target atoms. Electric arcs may be ignited causing an additional flux of atoms from the solid into the plasma. This book is divided into three parts. In Part I, fusion as a possible future energy source is shortly discussed, on the basis of the burn and exhaust criteria (Sect. 3.2), and the main fusion concepts, i.e., the magnetic and inertial plasma confinement, are outlined. The general design of the first wall in fusion experiments is considered. In Part II, the interaction of a plasma attached to a solid is discussed in terms of particle (Chap. 6), electric (Chap. 7), and power coupling (Chap. 8), after a chapter introducing the main plasma characteristics (Chap. 5). The main processes of plasma–surface interaction, as partly listed above, are described using rather simple analytical relations to show the parameters of importance and to make the underlying physics clear.1 In addition, numerical methods are presented for simulation of plasma–material interaction processes for the complex conditions usually met in experiments (Sect. 6.10, Sect. 7.6, Sect. 9.3.9). The serious problems caused by impurities in fusion plasma are described in Chap. 9. In Part III, a set of criteria, e.g., the impurity criteria (Chap. 10), as well as plasma operation limits (Chap. 11) and material operation limits (Chap. 12) are presented. The choice of materials for a future fusion reactor is critically discussed, and a summary of issues of concern and open questions is given (Chap. 13). 1

The SI unit system has been used throughout this compilation (energy and temperature are given also in eV)

Part I

Fusion as Energy Source

2 Energy Problem and Related Safety Aspects

The lifestyle of modern human civilization and the demographic development require an ever-increasing expenditure of energy. Today, the standard of living is directly linked to energy consumption, and there is little hope that this connection could be broken in the future. The global energy usage is increasing at about 2%1 p.a. resulting in an exponential growth over the years [10]. In dynamical systems that evolve under such conditions, sooner or later, saturation or collapse occurs due to exhaustion of resources. Currently, most electricity is provided by burning hydrocarbons or carbonbased fossil fuels. Smaller contributions are delivered by hydroelectricity, and by nuclear fission. Despite considerable financial encouragement, the renewable sources—wind, waves, direct solar heating, geothermal energy, photovoltaics—and a more efficient use of energy contribute only little on a global scale [11]. They all suffer from high costs, low energy densities, and low availability. On top of that, efficient use is possible only at remote locations far away from the centers of population and industry. For example, bioethanol is a possible substitute for petrol for road transport. The main energy crops used to produce bioethanol are sugar-beet and wheat crops. Lignocellulostic biomass (poplar and willow trees), waste straw and sawdust materials can also be used to produce bioethanol with a production rate of approximately 30 to 40 barrels (1 barrel = 159 l) per year and hectare. For comparison, the Organization of Petroleum Exporting Countries (OPEC) alone, which pumps about a third of the world’s crude supply, increased their daily oil output in 2005 to about 30 million barrels. In order to substitute the global daily amount of about 90 million barrels, one would need an area of about 25 000 km2 . To substitute the yearly production of oil, a farmland of 9 million km2 is required. The area available for agriculture is worldwide only about 10 million km2 and is almost completely used for food production. 1

In 2004 the increase of energy consumption exceeded the 4% level. The energy demand in China grew by 15% last year

6

2 Energy Problem and Related Safety Aspects

Today, a one-family house can be supplied with energy—with some financial effort—using photovoltaic sources, and it is easy to ignite the Olympic fire with the help of the sun and a concave mirror. However, providing industry, transport, and large cities with sufficient energy is up to now well beyond the means of renewable energy sources. It is feared that this situation will not improve in the near future. Efforts in energy saving and increased efficiency of energy systems in the technologically developed world are counteracted by the annual rise of energy consumption. It is a frightening scenario indeed, if the large population in developing countries such as China and India reach the world average of energy consumption. A massive increase in use of fossil fuels for power generation is going to take place in the developing world. China has already today the highest concentration of atmospheric sulfur dioxide. The consequences are not only the fact that fossil fuels, accumulated over millions and millions of years, are burned up in a few hundred years, but also acid rain, accumulation of CO2 in the atmosphere, and the associated greenhouse effect. Invaluable raw materials, which are subjected to an ever-increasing demand by the industry, among them the chemical and pharmaceutical industry, are reduced to simple molecules such as carbon oxides and water vapor. The ongoing depletion of world fuel resources has led and will increasingly lead to political and military confrontations. The use of fission power as a conceivable alternative is limited due to critical questions of safety, public acceptance, disposal of the radioactive waste, and proliferation of nuclear technology. Fusion power is a promising candidate, besides solar and fission power, for long-term energy supply. Each of these options should be developed to its full potential allowing coming generations to select the solution most suited for their needs. Since the fuel is fed on continuously in fusion devices (like fossil fuel plants but in contrast to fission reactors), the energy density in the reaction zone of a fusion reactor is comparably small and so is the hazard potential. Already small disturbances, e.g., a slightly higher impurity concentration, will lead to a quench of the burning fusion plasma. Chain reactions are not possible, since the fusion process is not based on neutron multiplication reactions. The deuterium fuel and the direct end product of fusion—the inert gas helium – are neither toxic nor radioactive. They do not produce atmospheric pollution, nor do they contribute to the greenhouse effect. In contrast, fission produces a multitude of highly radioactive elements. Nevertheless, a few kilograms of radioactive tritium will be on site of a fusion reactor. Its handling, however, will be facilitated by the fact that it is bred in the outer region of the reactor (in the blanket) and no transport over large distances is required. Studies have shown that routine losses of tritium to the biosphere can be made less than prescribed by regulatory guidelines, and that at maximum only a few hundred grams of the tritium inventory could be released under severe accident conditions [12].

2 Energy Problem and Related Safety Aspects

7

The neutrons from the fusion reactions are the main source of both penetrating radiation (requires shielding of primary and secondary neutrons and gamma rays) and radioactive contamination due to activation (containing and waste storage problem). The inventory of radioactive materials produced by neutron activation can be compared with the overall inventory in an equivalent pressurized water reactor. The toxicity, in terms of the gross potential to cause biological damage to man, is found to be significantly less than that in a fission reactor and many orders of magnitude less after a few years. The only significant source of afterheat, i.e., energy released from the decay of radioactive material, is associated with the structural material of the blanket, which should be cooled during both planned and emergency shutdowns to avoid damage. The use of low activation materials and carefully purified materials can reduce this inventory further, thus fully realizing the potential superiority of fusion over fission in this respect [12].

3 Fusion Fuel

In comparison to chemical reactions, where the released energies are in the range of a few eVs, fusion operates in the MeV range. That is why a rather small amount of fusion fuel produces a huge amount of energy, a million times more than in simple fuel combustion. The most competitive advantage of fusion is the almost inexhaustible fuel resource (Sect. 3.1). The problem is to initiate fusion, i.e., to overcome the repulsive force of the Coulomb interaction, and to sustain the fusion reactions on a longer time scale (Sect. 3.2).

3.1 Fusion Reactions Altogether more than 80 different fusion reactions are known. The fusion reaction with the largest cross-section at the lowest energy is the deuterium– tritium (DT) reaction D+T

→

4

He (3.517 MeV) + n (14.069 MeV) .

(3.1)

The released energy ∆E = ∆mc2 is due to the mass defect ∆m = mD+ + mT+ − (m4 He2+ + mn ) = 3.1 × 10−29 kg

(3.2)

according to the equivalence of mass and energy as result of Einstein’s special theory of relativity. Only this fusion reaction is being regarded for a first fusion reactor. Its effective cross-section (T < 100 keV [13]) ln(σvDT ) [cm3 /s] =

−21.38 − 25.2 − 7.1 · 10−2 Te + 1.94 · 10−4 Te2 Te0.29 +4.92 · 10−6 Te3 − 3.98 · 10−8 Te4

(3.3)

(for T = 23 keV σvDT = 5.14·10−22 m3 /s) is about 100 times larger, between 1 and 100 keV, than that of the deuterium–deuterium (DD) reactions

10

3 Fusion Fuel

D+D D+D

→ →

3

He (0.817 MeV) + n (2.450 MeV) T (1.008 MeV) + p (3.024 MeV) .

(3.4) (3.5)

The temperature in (3.3) is given in keV. The simple relation [14] σvDT = 1.1 × 10−24 (T [keV]) m3 /s 2

(3.6)

is valid near T = 10 keV. More exact formulae, valid in a wider range of temperatures, can be found in [15]. On the other hand, the neutrons produced in the DD reactions are less energetic and therefore less harmful to the reactor structure materials. However, in initial pure D plasmas tritium will also be produced, and therefore, so will highly energetic neutrons according to (3.1). The energy of the α-particles (helium nucleus) is transferred by collisions to the fuel, i.e., to D and T, thus helping to establish a self-sustaining reaction. The fast neutrons easily penetrate the fuel zone and have to be slowed down in a specially designed wall element (blanket), where their energy is converted into heat and then into electricity applying conventional water–steam technology. High energies of the reacting particles are required to overcome the Coulomb barrier ( 0.4 MeV), but, thanks to the tunnel effect [16], not as high as estimated from classical calculations. The probability for a particle of mass m and velocity v to cross a potential barrier can be roughly estimated by
 ∆ (3.7) fG = exp − ∆λ where for the characteristic decay length ∆λ , the de Broglie wavelength ∆λ = h/(mv) of the incoming particle can be taken. The width of the barrier ∆, i.e., the distance to cross the barrier, is obtained by calculating the distance between two particles with their nuclear charges Z1 and Z2 at which the potential energy is equal to the kinetic energy of the incoming particle leading to Z1 Z2 e2 mv 2 Z1 Z2 e2 = → ∆= . (3.8) 4πo ∆ 2 2πo mv 2 Using this expression, the tunnel probability is proportional to
 Z1 Z2 e2 fG ∝ exp − o hv

(3.9)

an expression close to the famous Gamov factor exp(−πZ1 Z2 e2 /(o hv)) [16]. Without the tunnel effect, plasma temperatures of several billion K would be necessary in comparison to the significantly lower temperatures of several 100 million K envisaged in fusion reactors. Deuterium, a non-radioactive isotope of hydrogen, is very abundant and may be extracted from seawater. Deuterium is found in natural hydrogen

3.1 Fusion Reactions

11

compounds to the extent of 0.014 to 0.015 percent, i.e., roughly 1 D in every 6500 H atoms. The estimated resource in the oceans is 4.6×1013 tons, sufficient for ten billion years based on today’s average annual world consumption [17]. It is possible to use the different boiling points of heavy water (101.4o C) and normal water (100o C) for separation, but this would require a lot of energy due to the high heat of vaporization of water. More efficient methods include distillation of liquid hydrogen and various chemical exchange processes which exploit different affinities of deuterium and hydrogen for various compounds. These include the ammonia/hydrogen system, which uses potassium amide as the catalyst, and the hydrogen sulfide/water system. Tritium is an isotope with a relatively short period of radioactivity (halflife of 12.3 years), so it does not occur in nature and should be bred on site from lithium via either the reaction 6

Li + n

→

4

He (2.05 MeV) + T (2.73 MeV) ,

(3.10)

which can be achieved by slow neutrons, or via the reaction 7

Li + n

→

4

He + T + n − 2.47 MeV

(3.11)

using fast neutrons. The latter reaction does not consume a neutron. Each fusion neutron could produce one new tritium atom. Lithium-6 makes up 7.4% of natural lithium. The average quantity of lithium in the Earth’s crust is around 50 ppm. It is more abundant than tin or lead and even ten times more abundant than uranium (3 to 4 ppm) [17]. Reserves of Li are estimated at 12 million tons, if taken from seawater, an additional potential of 230 000 million tons would be available. A self-cooled lithium blanket appears to be the simplest and most reliable approach to a breeding blanket in a reactor. In order to balance losses in the blanket, a small additional neutron source, e.g., via the reaction 9

Be + n

→

24 He + 2n − 1.57 MeV

(3.12)

is necessary since only one neutron is produced in each fusion reaction. In the new ITER design such a breeding blanket is not foreseen [5, 18]. The basic function of the planned blanket system is to provide thermal and nuclear shielding to the vessel and external components. A partial conversion of this shielding blanket to a breeding one is envisaged for a later state of operation.1 The fuel resources, limited by lithium, are estimated at several thousand years if the lithium is of telluric origin and at several million years if the lithium is taken from seawater. The realization of fusion power on earth would mean a virtually limitless source of energy at comparatively low fuel cost. The enormous impact this would have on our civilization makes controlled fusion the most important scientific challenge man has ever faced [19]. 1

see also http://www.iter.org

12

3 Fusion Fuel

Apart from fusion fuel other materials needed for construction of thermonuclear reactors might run out. A fusion reactor consists of a large number of high-technology components containing tons of rare, non-ferrous materials. For example, the world production capacity for superconducting wires such as NbTi (Nb3 Sn and MgB2 wires are the newest superconducting material under development) will be almost fully occupied by providing just the next fusion device ITER. The extensive use of high-temperature superconductors (HTS) based on rather rare materials such as yttrium, bismuth, thallium or mercury is therefore also questionable, not only because of their structural problems. Carbon fiber composites (CFC), ceramic fiber matrices and other advanced materials requiring an extraordinary technological effort would be needed in bulk. Helium is a unique industrial gas. However, only a handful of sources in the world produces it. The high cost of extraction restricts helium use to relatively few high-technology applications. By far the largest use of helium is as liquid coolant for various superconductors. This use accounts for about 30% of global demand. There is growing concern over the ability to meet future supply needs. Hydrogen has one major advantage over helium in that it is freely available at relatively low cost. Hydrogen could serve equally well for many helium applications, but it is flammable, therefore explosive. And though hydrogen’s boiling point is only a few K higher than helium, it cannot be used as a cheaper substitute for cooling low-temperature superconductors.

3.2 Ignition and Burn Criteria For a continuously burning DT fusion plasma, appropriate parameters (particle and energy confinement times, plasma density, and temperature) have to be achieved in order to be able to exhaust the produced 4 He and the energy [20, 21]. Fulfilling the particle and energy balances, only a limited operational window for a fusion reactor is available [22–24], which is even more narrow due to plasma impurities and operation limits such as the β and density limits (see Chap. 11). In the following, the concentrations of helium and impurity ions are denoted by fHe = nHe /ne and fi = ni /ne , respectively, relating the corresponding densities to the electron density. Taking charge neutrality   qi ni = nD + nT + 2nHe + qi ni = ne (3.13) i

and equal densities of deuterium and tritium, nD = nT = nDT , we obtain   ne  (3.14) nDT = 1 − 2fHe − qi fi 2 where qi is the charge state of ions of kind i and fi is their concentration (fi = ni /ne ). In particular, qi is the charge state of impurity species i and

3.2 Ignition and Burn Criteria

13

qHe = 2. The dilution of the DT fusion fuel by helium  and impurity ions is characterized by the additional factor (1 − 2fHe − qi fi ), which reduces the fusion rate. Under steady state conditions, the fusion rate (equalizing the production rate of α-particles) must balance the exhaust rate of helium nD nT σvDT =

nHe ∗ τHe

(3.15)

where σvDT is the fusion rate coefficient. The particle confinement time, τHe , is defined by  nHe dV τHe = volume (3.16) Γ dS surface He where Γ is the helium flux density out of the volume V enfolded by the surface S. The He exhaust itself can be parameterized by introducing the effective confinement time [25] τHe τHe ∗ τHe (3.17) = = 1 − Rcyc  where Rcyc is the recycling coefficient and  = 1−Rcyc is the exhaust efficiency determined by the scrape-off layer (SOL) physics and pumping capabilities. During a typical discharge, a helium particle leaves and re-enters the plasma many times before it can be pumped out. The effective confinement time is ∗  τHe , usually considerably longer than the particle confinement time, τHe since Rcyc is close to unity (Fig. 3.1). When external fuel sources such as a gas inlet are turned off, the density is observed to decay with a characteristic time τp∗ , i.e., as exp(−t/τp∗ ) [2], since edge fueling by recycling continues and the confinement/replacement time is given by (3.18) τp = (1 − Rcyc )τp∗ . The difference between τp and τp∗ becomes critical by analyzing the burn criterion (see below). In fusion experiments τp is usually on the order of tens of milliseconds, while τp∗ is about one second. Central Plasma Particles Energy

1

+

R cyc

+

2

R cyc

+

3

R cyc

+

...

=

1 1 - R cyc

Plasma Facing Component

Fig. 3.1. The recycling process. The energy is almost fully absorbed by the material, whereas particles such as He can leave the central plasma region and return many times

14

3 Fusion Fuel

For a fusion reactor, the produced helium ash corresponding to the produced fusion power has to be removed as well as the additional fueling in the case of neutral beam injection (NBI) of highly energetic deuterium, if it is used as a plasma heating method. Unfortunately, the pumping system will probably exhaust the hydrogen isotopes with an efficiency comparable to that of helium. Thus, the required throughput of helium will be accompanied by a large throughput of DT particles due to the lower concentration. The pumping requirements are therefore quite substantial. Combining (3.14) and (3.15) gives the so-called exhaust criterion ∗ = ne τHe

4fHe .  2 (1 − 2fHe − qi fi ) σvDT

(3.19)

The plasma is transparent for the highly energetic neutrons produced in the fusion reactions. They penetrate more or less freely through the first wall with its thickness of a few centimeters. Their kinetic energy will be converted into thermal energy in thick blanket structures located behind the wall. Contrarily, the part of helium fusion power Pα (Eα = 3.52 MeV, see (3.1)) which is not lost through radiation must be confined in the central plasma long enough to replace convective and diffusive heat losses in order to maintain the plasma temperature WE 3  ne kB Te = Pα − Pbrems − Prad = τE 2 +2nDT kB TDT + nHe kB THe (3.20)   τE + ni kB Ti where WE is the energy content of the plasma, τE is the energy confinement time, Pbrems are the power losses due to bremsstrahlung and Prad are the additional power losses due to impurity radiation. In striking contrast to particle confinement, τE is predominantly determined by transport processes, since most of the energy leaving the plasma is not recycled back from the wall. The helium fusion power is given by 2  n2  fi qi σvDT Eα . (3.21) Pα = n2DT σvDT Eα = e 1 − 2fHe − 4 Since fully ionized ions do not radiate (besides bremsstrahlung), neither DT nor helium contribute to Prad , but only partially stripped impurity ions. Recombination can be neglected in the hot central plasma. Introducing radiation functions Li , which are functions of electron temperature Te and charge state of the impurity ion species i, Prad becomes  Prad = ne (3.22) ni Li (Te ) .

3.2 Ignition and Burn Criteria

15

Pbrems can be written (see (9.9)) as (with qDT = 1 and qHe = 2)
  qi2 ni kB Te = cbr n2e Zeff kB Te Pbrems = cbr ne 2nDT + 4nHe + 

i

  1 + 2fHe + qi2 − qi fi kB Te  since 2nDT /ne = 1 − 2fHe − qi fi (see (3.14)), and √ 16 2πγG = 3.84 × 10−29 Wm2 s/ kg cbr = √ 3/2 3 3(4πo )3 me c3 ¯h =

cbr n2e



(3.23)

(3.24)

where γG  1 is the Gaunt factor, which was introduced by Kramers to take account for discrepancies between the classical and quantum mechanical treatments. Zeff is the effective charge state of the plasma ions  2 qi ni   all ions Zeff =  = qi2 ni ne . (3.25) qi ni all ions

all ions

As far as bremsstrahlung is concerned, in an ignited DT fusion reactor, medium and light impurities can be tolerated up to an effective space charge of about 2, whereas for DD and DHe3 plasma ignition is possible only for very clean plasmas [23]. Substituting (3.21), (3.22), and (3.23) into (3.20), we obtain with the assumption of T = Te = TDT = THe = Ti    ne τE = (3/2) kB T 2 − fHe + fi [1 − qi ]  2  1 − 2fHe − fi qi σvDT Eα /4

 

   2 qi − qi fi kB T − fi Li . (3.26) −cbr 1 + 2fHe + The exhaust criterion (3.19) and the burn criterion (3.26) can be combined in order to eliminate the electron density by defining the ratio [22] γ=

∗ τHe , τE

(3.27)

which results in a cubic equation for a certain impurity concentration fi for given γ, T , and fHe 2   8fHe  1 − 2fHe − γ= fi qi σvDT Eα /4 − cbr 1 + 2fHe 3

  2 

    2 1 − 2fHe − + qi − qi fi kB T − fi Li qi fi

   × 2 − fHe + fi [1 − qi ] σvDT kB T . (3.28)

16

3 Fusion Fuel

Fig. 3.2. Operation space of a fusion reactor considering only helium as an impurity, according to (3.29). The chosen values of γ are indicated. Here, the fusion rate coefficient is taken according to (3.3)

In the case of fi = 0 (only helium, no other impurities), this reduces to √ 2fHe σvDT Eα − 4cbr kB T (1 + 2fHe )/(1 − 2fHe )2 γ= , (3.29) 3 (2 − fHe ) σvDT kB T which is also a cubic equation in regard to fHe . Due to charge neutrality 2nDT + 2nHe + qimp nimp = ne

(3.30)

(here only one impurity species with charge state qimp is considered) an additional condition arises fimp =

nimp 1 − 2fHe ≤ ne qimp

(3.31)

reflecting the extreme case of nDT = 0 in the case of the equal sign. In a plasma without any impurities, other than helium, the limiting condition is fHe ≤ 1/2. Usually, (3.29) has two real solutions, fHe,1 and fHe,2 , for a given γ value and a certain temperature in the available range of 0 ≤ fHe ≤ 1/2. Varying the temperature T , the obtained solutions of (3.29) can be substituted into the burn criterion (3.26). The resulting values of ne τE build the workspace of a fusion reactor as a function of T for a certain value of γ (Fig. 3.2).2 As seen, there are no solutions at very low temperatures. A real point of concern is the fact that the operation space is tied up with increasing 2

Often, a similar presentation by means of the so-called triple product ne τE Ti as a function of temperature is used in the literature [22, 24]

3.2 Ignition and Burn Criteria

17

values of γ. For γ ≥ 16, there is no solution at all (Fig. 3.2). Assuming linked transport mechanisms for particle and energy transport, i.e., τHe  τE , the exhaust efficiency  (see (3.17) and (3.27) should be larger than  > 1/16  0.06. Confinement times are strongly dependent on the source location (the sink is assumed to be fixed at the edge). The values for τHe and τE can be quite different, even if the transport coefficients are numerically the same, since energy sources are usually in the plasma center while particle sources (neutral ionization from recycled particles) are located near the edge. ∗ /τE was measured to be approximately 8 in the fusion The ratio γ = τHe experiment DIII-D, using helium gas puffing as well as 75 keV neutral helium beam injection simulating a central source of helium [26]. In ASDEX-Upgrade, a ratio of about 7 has been demonstrated in helium exhaust studies [27, 28]. From the statistical analysis of confinement results obtained in different fusion devices, expressions of the energy confinement time τE are found as a function of geometrical and physical parameters. The expression [18] 0.15 −0.69 0.41 τEtokamak = 0.0562 Ip0.93 BT P ne M 0.19 R1.97 ε0.58 κ0.78 [s]

(3.32)

is valid in tokamaks with H-Mode and ELMs. The parameters and units are the plasma current Ip in MA, the toroidal magnetic field BT in T, the total power crossing the separatrix P in MW, the electron density in 1019 1/m3 , the average ion mass M in amu, the major radius R in m, the inverse aspect ratio ε = a/R, and the elongation κ. For stellarators, the following relation is proposed [29] 0.88 −0.77 0.69 0.04 P ne ι [s] τEstellarator = 0.048 a2.39 R1.22 BT

(3.33)

with the rotational transform ι = 2π/qs and the minor radius a. The parameters are given here with the same units as in (3.32). The energy confinement time is usually calculated from the measured plasma energy WE and heating power P according to W . (3.34) τE = P − dW/dt

4 Fusion Concepts

Although the sun successfully demonstrates that fusion works, its confinement principle cannot be applied, since gravitational forces are far too weak for terrestrial plasmas. There are two basically different approaches to controlled thermonuclear fusion on our planet: magnetic and inertial confinement fusion.

4.1 Inertial Plasma Confinement In the latter method, the surface of a small pellet containing the fusion fuel is rapidly heated by high-energy lasers or particle beams. By a rocket-like inward reaction the pellet implodes. The fusion fuel is compressed to super high densities and is adiabatically heated until the pellet core is brought to ignition. The confinement time needed to comply with the (ne τE )-criterion (see Sect. 3.2) and should be smaller than the time during which the inertia of the pellet material is sufficient to hold the pellet together (inertial confinement). In the direct drive approach, several beams are producing a highly symmetric illumination with an optimized pulse shape. Using an indirect drive, the beam radiation is first converted to soft x-rays that isotropically fill a cavity containing in its center the fuel pellet. The key issues of the inertial confinement fusion are the pellet design, the efficiency and repetition rate of the laser or particle beams, and the cyclic energy deposition. Additional problems arise during the repetitive ignition of pellets one after the other falling down into the reaction vessel. Already after the first two or three successful ignitions, the resulting “dirty” conditions in the reaction vessel would not allow a further localized energy input with the required high accuracy. The long cleaning phases between the ignitions and the required reloading time of the drivers are the problems to encounter on the way to, at least, quasistationary conditions [30–32]. Plasma and particle interaction processes with the wall are quite similar to those in the magnetic confinement concept (see below), while being here of less concern [33].

20

4 Fusion Concepts

Laser physicists in Europe have proposed a new “fast ignition” facility HiPER. In the conventional approach to inertial confinement the lasers that compress the fuel pellet also heat it. Fast ignition relies on the usage of different lasers for these two phases. The HiPER facility would consist of a long-pulse laser with an energy of 200 kJ for compression and a short-pulse laser with an energy of 70 kJ for heating [34].

4.2 Magnetic Plasma Confinement In magnetic confinement, the currents of the charged particles, ions and electrons, are impeded from flowing across the magnetic surfaces, i.e., in the radial direction. Soon after starting with linear devices, which had a mirror configuration of the magnetic field coils, it became clear that the magnetic field lines have to be bent into a torus in order to avoid the significant parallel losses. To provide toroidal plasma equilibrium and stability, the magnetic field lines have to pass around both the poloidal way and the toroidal way to form magnetic surfaces [14, 19, 35, 36]. Without the poloidal field, the magnetic field gradient would cause charge separation between the top and bottom of the plasma. The resulting electric field would drive the plasma outwards to the wall. The helical structure of the magnetic field configuration gives stability by shorting out charge imbalances. The amount of the helical twist is given by the so-called safety factor qs , which becomes approximately qs 

r Btoroidal R Bpoloidal

(4.1)

in the case of circular cross-section plasmas, where r and R are the minor and major radius, respectively. Since the temperature for ignition is defined and the reacting plasma must have a sufficiently low pressure (see below, β-criterion), the plasma density is rather low (seven to eight orders of magnitude lower than the particle density in solids). In order to fulfill the (n τ )-criterion (see Sect. 3.2) the time τ , which characterizes the energy confinement, should typically be a few seconds. The most promising concepts of magnetic confinement fusion are the tokamak and stellarator systems.

4.3 Stellarator Concept In the stellarator, initiated in 1950 by Spitzer, the vacuum magnetic field for plasma confinement is produced entirely by currents flowing in external coils (Fig. 4.1). In the absence of a net toroidal plasma current, the magnetic configuration can be maintained in steady state provided only that currents in the coils are maintained and that particle and energy exhaust can be ensured

4.3 Stellarator Concept

21

Planar Coils

Non-planar Coils Magnetic Field Line

Plasma

Fig. 4.1. Schematic of a five-period optimized stellarator with modular coils

together with refueling. No external current drive system is needed. The system is necessarily asymmetric in the toroidal direction. Despite the increased engineering complexity, stellarators do not show disruptive instabilities due to a quench of the plasma current as can happen in tokamaks, and this could prove decisive in the long term. The absence of a net plasma current as in tokamaks minimizes the free energy available for driving instabilities. With the Wendelstein line, the Max Planck Institute for Plasma Physics (IPP) is exploring the reactor potential of advanced stellarators [37–40], a concept for confining toroidal plasmas with magnetic fields generated exclusively by a set of external, planar, and non-planar coils. Progress in experiments and theory has eliminated major concerns about low stability limits and enhanced transport losses due to the large magnetic ripple in stellarators. The engineering concept of modular coils [37] allows the realization of an optimized magnetic configuration with considerable reactor potential. As a consequence of optimization, the configuration possesses an inherent magnetic separatrix with three-dimensional particle and energy fluxes at the plasma boundary. The three-dimensional nature of the boundary topology gives rise to a rather complex plasma–wall interaction pattern. Transport simulations [41] have shown that the plasma outflow is rather concentrated in regions where the flux surfaces show the strongest curvature in the poloidal cross-section. This region corresponds to a helical ridge, which starts at the lower apex of the elliptical cross-section and stretches along the outside to the upper elliptical apex through each field period [42, 43]. Correspondingly, visible ribbons of deposition have been observed along the vessel wall [44]. The development of a proper exhaust technology is another important task. One of the interesting properties of the optimized design is the adapta-

22

4 Fusion Concepts

tion to divertor operation (see below), important with respect to power and particle exhaust, impurity control, and minimization of erosion and impurity production. The exhaust will depend on the island structures at the edge of the plasma and the degree of ergodization there [45]. The aspect ratio, which is the ratio of the major plasma radius to the minor plasma radius, is larger than that of tokamaks, hence, the power loading onto the first wall is correspondingly reduced. On the other hand, a fusion reactor based on the stellarator concept will be large with a radius of about 25 m. The use of superconducting magnetic coils, for example using NbTi conductors, is inevitable. Without net toroidal current, heating relies upon non-ohmic methods such as electron and ion cyclotron resonance heating (ECRH, ICRH) and neutral beam injection (NBI).

4.4 Tokamak Concept In the tokamak, proposed by Sakharov in 1952, the poloidal magnetic field component is produced by a toroidal current, flowing in the plasma itself. This current is induced by transformer action. The primary winding changes the magnetic flux and the resulting toroidal electric field with a loop voltage of a few volts drives the plasma current. Furthermore, the plasma is ohmically heated by the induced current. The toroidal field is produced by currents flowing in uniformly distributed poloidal coils encircling the plasma torus. As a result, the plasma is free to flow on certain closed and nested surfaces—the axis-symmetric magnetic surfaces (Fig. 4.2). If a tokamak plasma is heated only by ohmic heating, the ohmic heating power density ηj 2 decreases and the bremsstrahlung loss increases with increasing temperature. Therefore, the upper limit of the electron temperature can be estimated from ηj 2 = Pbrems . This limit is less than a few keV. Additional heating, e.g., neutral beam heating, is necessary in a tokamak system to approach the condition for ignition. The tokamak confinement system has been well developed and experiments demonstrated high fusion triple products: ne τE TDT (Sect. 3.2). With the ITER tokamak, the upcoming project of international cooperation, the scientific and technological feasibility of fusion energy should be demonstrated using the magnetic confinement concept [18]. The device is conceived for controlled ignition and extended burn of DT plasmas. During the design engineering activities (EDA phase, starting in 1992), its technical concept has been developed and progressively evolved fulfilling the revised programmatic objectives and the cost reduction target adopted by the ITER parties in 1998. While having reduced technical objectives, the new ITER design (ITER–FEAT for Fusion Energy Advanced Tokamak) will provide an integrated demonstration of a burning fusion plasma addressing confinement,

4.5 Design of the First Wall

23

Transformer Coils (inner poloidal field coils)

Outer Poloidal Field Coils

Toroidal Field Coils Magnetic Field Line

Plasma Plasma Current

Fig. 4.2. Schematic of a tokamak. The transformer induces the toroidal current. The outer poloidal field coils are used for plasma positioning and shaping

stability, exhaust of helium ash, and impurity control as well as the availability of essential fusion technologies. In 2005, the site was selected. ITER is to be constructed in France, at Cadarache. After the construction time of about ten years, the device is anticipated to operate over an approximately 20-year period in order to execute its challenging program [18]. ITER is also an experiment for investigating plasma–wall interaction issues to prepare a viable solution for future steady state reactors such as DEMO.

4.5 Design of the First Wall Due to the limited confinement of energy and particles and the demands connected with the aim to convert the thermal energy into electricity in a future fusion reactor, the design of the first wall elements requires special care and attention. It has to perform three main functions: (1) to sustain the impact of the energetic particles and of radiation without releasing many impurities and without a large degradation of its thermophysical properties, (2) to transfer the heat load of about 0.5 to 1 MW/m2 coming from the plasma to a cooling medium and to withstand the resulting thermal stress, while allowing the neutron flux to reach the blanket modules where the kinetic energy of the neutrons is used to heat up a coolant as in a primary loop of a conventional heat-power plant, (3) be able to withstand high heat loads

24

4 Fusion Concepts

during events such as disruptions, ELMs or generation of runaway electrons and the related electromagnetic loads, (4) minimizing the tritium retention in accordance with the general objective of a low tritium inventory. The crucial problem is to save power exhaust from the core plasma without intolerable effects to the first wall on one side and to the plasma itself on the other. Radiation at the edge can help to transfer the power onto sufficiently large areas, as well as the power distribution over sufficiently many particles by recycling. Impurity sources from the wall can largely contribute to the contamination of the core plasma because of the large wall area and the ineffective redeposition of once emitted particles. The sources of impurities can also be very localized as observed, for example, at the limiters of ICRH antennae. One can try to restrict the area of plasma–wall interaction (Chap. 6) to dedicated regions by exposing so-called limiters into the plasma or to divert the edge plasma by means of special magnetic coils into a so-called divertor. In both schemes, the plasma is streaming along the magnetic field lines towards the exposed surfaces, where it is neutralized. Most of the plasma electrons are expelled by the electric field being automatically established in the thin sheath above surface—the electric sheath (Chap. 7). The plasma ions are accelerated up the ion sound speed by reaching the sheath entrance and recombine at the surface. To ensure the equality of ion and electron fluxes, the surface is negatively charged with respect to the plasma potential. The loss of plasma is partly compensated by dissociation and subsequent ionization of emitted molecules but mainly balanced by the diffusive transport out of the undisturbed core region into the edge plasma. The thickness of the so-called scrape-off layer (SOL) formed by the sink action of limiters or divertor plates (Fig. 4.3) can easily be estimated by equating the cross-field particle flux into the SOL of length Lc and poloidal extent l [46, 47]  dn  nLCFS D⊥  Lc l ≈ D⊥ Lc l (4.2) dr  λSOL LCFS

to the particle flux reaching a wall element wall 

l

ne (r)cs (r) dr ≈ 0.5 nLCFS cs λSOL l

(4.3)

LCFS

where Lc is the shortest connection length to an absorbing wall element (limiter, divertor, or wall) and cs = kB (Te + Ti )/mi is the isothermal ion acoustic velocity. The plasma density nLCFS far from the wall element is reduced roughly to half of the value near above the surface. The cross-field diffusion in the boundary layer, characterized by the diffusion coefficient D⊥  1 m2 /s, is slow enough in comparison with the parallel motion to give rise to a sharp plasma boundary with a short plasma decay length

4.5 Design of the First Wall

25

Separatrix

SOL Core Plasma

Limiter Core Plasma X-Point

SOL

LCFS Private Flux Region

Divertor

Fig. 4.3. Limiter and divertor configuration shown in the poloidal cross-section of a tokamak with indicated flow pattern

 λSOL =

2D⊥ Lc = 2D⊥ τSOL cs

(4.4)

on the order of several centimeters. It is worth noting that an exponential decay of the plasma parameters in the scrape-off layer can also be explained, in another way, by an increasing cross-field transport across the SOL, for example, by increasing D⊥ with increasing radius. The extreme anisotropy of heat and particle transport due to the dominant parallel flows in the SOL lead to very localized regions of high load onto the limiter or divertor targets. This constraint can only be faced by establishing an extremely glancing incidence of the magnetic field with an angle of about one degree. However in practice, wall tolerances and magnetic error fields cause a variation of the local angle typically of the same order. For such conditions, sweeping of the magnetic field configuration in the divertor, i.e., changing the strike point position of the separatrix at the divertor plates in time, could spread the power deposition over larger areas. 4.5.1 Limiter Limiters are very useful for concentrating the plasma surface interaction on specially designed and geometrically optimized structures (Fig. 4.3). The main disadvantage of limiters is the high possibility that eroded particles once ionized will reach the central plasma by cross-field diffusion, since ionization will probably occur already inside the confined region. Even a proper alignment, i.e., establishing an almost parallel incidence of the magnetic field lines at the top of the limiter, would not reduce the particle flux density to these areas (and subsequently the erosion from there) down to zero as one would hope applying the simply geometrical sin α-law,

26

4 Fusion Concepts

where α is the angle between the magnetic field line and the surface. The substantial heat and particle fluxes are dominated in this case by direct, crossfield transport [48]. However, using strong gas injection, for example neon, a cold radiative plasma mantle could be established in TEXTOR [49,50]. Such a concept allows for dissipation of the energy on the whole plasma vessel, thereby reducing the load onto the limiter. Nevertheless, the crucial limitation for radiative edge cooling is also given by the impurity concentration in the center. 4.5.2 Divertor With additional magnetic coils, usually placed at the top and/or bottom of tokamaks, the magnetic topology in the plasma boundary is changed in a way that previously closed magnetic flux surfaces are open and particles, which pass beyond the separatrix, are directed to target plates far away from the hot central region (Fig. 4.3). Although the radial distance of the divertor region to the separatrix is usually less than a meter, the particles have to move a long distance up to hundreds of meters along the magnetic field lines before reaching the targets. This opens the possibility to actively control the plasma parameters in the divertor region without affecting the plasma core parameters and, therefore, the confinement. The divertor should decouple the plasma parameter at the edge from those at the target plates [51–53]. It has already been pointed out earlier by D¨ uchs et al. [54] that steady state operation is not feasible without a divertor when considering that the following goals have to be achieved: (1) to exhaust the power at an acceptable erosion rate, (2) to control the impurity content of the main plasma by retention of impurities produced at the divertor targets, (3) to pump the He ash out of the confined region. Given that particle and energy fluxes can be transferred to the wall components either at a high temperature and low plasma density (called low recycling regime) or at low temperature and high density (high recycling). The most promising divertor concept seems to be the losses-in-series using the combination of all available energy loss mechanisms from the core region down to the divertor to disperse the power by atomic processes as the only way to reduce the heat load onto the plates [55, 56]. The power loss channels are: (1) impurity radiation in the SOL and in the divertor, (2) momentum and energy transfer by charge-exchange (and elastic ion-neutral collisions) neutrals to the divertor chamber wall, a concept called “neutral cushion”, characterized by a flame-like envelope of the ionization front (Te  5 eV), (3) volume recombination by three-body recombination, which becomes effective at low plasma temperatures (< 2 eV), and (4) ionization and hydrogen radiation near the targets, i.e., high recycling. Impurity radiation mainly occurs in regions where the electron temperature is about 10 eV and higher, hydrogen ionization occurs near 5 eV, and plasma recombination at 1 eV and lower. Only by employing all of the listed effects, e.g., using gas puffs and

4.5 Design of the First Wall

27

feedback-controlled injection of recycling impurities such as Ar or Ne [50], a reduction of the heat load down to an acceptable value of 5 to 10 MW/m2 can be realized. Such additional measures as the increase of “wetted area”, by tilting the target poloidally (which is limited due to alignment and plasma position control) and the spatial sweeping of strike zones (which needs space and makes tight baffling of neutrals difficult) could only help to some degree but not solve the problem. There is no material and/or target design in sight to stand a higher load under steady state conditions. Thus, a successful divertor design is based on the optimization of both sides of the problem: an adequate choice of the material and optimized plasma regimes. Reducing the plasma temperature at the target plates offers the favorable possibility of decreasing the erosion yield. For some materials, e.g., high-Z elements, the energy of impinging ions could then be even smaller than the threshold energy for sputtering. For the same energy flux, it is much better— with respect to erosion—to have high densities and low temperature near the target plates. The primary indication of such a detachment is a decrease up to an order of magnitude in the ion saturation current, recorded, for example, by Langmuir probes embedded in the target plate. Released impurities will be ionized near the plates. Their parallel streaming is effected by (1) friction forces, which tend to bring their flow in-line with the streaming velocity of the background plasma ions, by (2) thermal forces ∝ ∂Ti /∂s which tend to drive the impurities back from the plates, and by (3) electric forces which occur because of the electric potentials needed to establish ambipolarity of ion and electron fluxes. In particular the thermal forces imply a sufficient plasma flow towards the divertor plates that drives the impurity ions back to their place of birth. The ratio of the probabilities for impurities to enter the main plasma, either when emitted from the divertor plates, or when they are sputtered at the main chamber wall allows for the assessment of the retention capability of the divertors [57]. Especially under high-density conditions, where the re-ionization of the fuel neutrals is close to the target, the unfortunate case can occur that certain flux tubes will experience an excess of ionization, resulting in flow reversal in that flux tube. In such a case, the number of fuel neutrals ionized in the tube exceeds the outflow of plasma (fuel) to the plate [58–60]. Not only is the retention of impurities important, but also the fuel neutrals should be kept inside the divertor regions by means of a closed divertor design, which is essential in order to allow efficient pumping of H and He and to prevent charge-exchange sputtering at the main plasma wall. The neutrals must recirculate in the whole divertor chamber. Efficient momentum removal and thermalization of hot neutrals require the wall to be as close as possible to the plasma. Possible solutions are a wide gas box with transparent walls close to the plasma or, as a possible alternative, the application of vertical targets in the “slot divertor” [61]. In particular, charge-exchange neutrals are

28

4 Fusion Concepts

only helpful if they can escape, travel to a region of high ion temperature, and return as cold ions to the target. In a slot divertor, the neutrals transfer their energy to the side walls. If they are ionized in front of the target, they will give their energy to the same local region and only marginally contribute to a reduction of power flux densities. The He pump capability of the divertor depends, on one hand, whether it will be possible to achieve a sufficient accumulation of He in the divertor plasma, and on the other hand, requires a pump which recycles the hydrogen isotopes such as D, T, and retains He. In addition to the normal load, the divertor components have to withstand electromagnetic loads due to disruptions leading to eddy currents up to several MA conducted through the divertor structure and resulting in loads of several hundred tons. A special divertor concept has been developed for stellarators and will be tested in future experiments. The complex plasma geometry characterized by island structures at the edge offers relaxed possibilities to enable the required particle and power exhaust. It has been shown that in the case of optimized stellarators the critical “leading-edge” problem, i.e., excessive heat and particle deposition due to vertical impact of field lines on the targets, can be solved [45] in spite of the complex three-dimensional geometry.

Part II

The Plasma-Material Interface

5 The Plasma State

The equilibrium between the thermal (or random kinetic) energy of the particles and the interparticle binding potential determines the state of matter. A plasma is a quasineutral gas of charged and neutral particles, while the transition from a gas to a plasma occurs gradually with increasing temperature. In a neutral gas, all macroscopic forces are transmitted to each particle by collisions. A plasma exhibits in addition to binary collisions collective behavior, i.e., the motion of charged particles generates electric fields and currents, which, in turn, give rise to magnetic fields. Owing to the particle motion in these self-generated fields (and external fields) the plasma is able to sustain a great variety of wave phenomena and show its unique property: long-range interaction. The term plasma is used to describe media containing free charged particles, which remain macroscopically neutral. Three conditions must be satisfied to call an ionized gas a plasma: 1. 2. 3.

The Debye length λD (Sect. 5.2) must be smaller than the plasma dimension. The number of particles ND in a sphere with radius of λD should be large in ideal plasmas. The frequency of typical plasma oscillations ωpl (Sect. 5.3) should be larger than the reciprocal value of the mean time between collisions with neutral atoms.

The three parameters, λD , ND , and ωpl , which characterize a plasma and its behavior, are in close relationship to each other [62, 63]. In particular, a usually several tens of thousands, over shift of ND = ne (4/3)πλ3D electrons, a distance of about λD = o kB Te /(ne e2 ) (keeping the positively-charged particles at their initial positions) leads to an increase of potential energy in that region, which is comparable to the kinetic energy of one electron ∝ kB Te . The plasma frequency ωpl is roughly the reciprocal value of the time

32

5 The Plasma State

tpl required for an electron to fly over a distance of λD  kB Te /me ne e2 1 ve = ωpl  = ≈ tpl λD o me o kB Te /(ne e2 )

(5.1)

and defines how fast a plasma (and especially the electrons) can react on disturbances of charge neutrality. The extent of disturbed regions, i.e., local regions where charge imbalance is allowed in a plasma, is given by the Debye length. Every forced disturbance is met by the plasma with an adequate rearrangement of the charged particles. This collective adjustment is much faster and more effective than the momentum and energy change due to binary collisions. In fact, the so-called self-collision time τc (see (5.50)), a measure of the collisions between charged particles, is much larger than τpl = 1/ωpl tc 2π2 m2 v 3 ve = 4 o e e = tpl e ne ln Λ λD




o (me ve2 )/3 e2 ne

2

9·2 3 9 · 2π n = πλ ne ≈ ND (5.2) λD ln Λ ln Λ D

since ND is a large number in a low-density, ideal plasma. The collective behavior of plasmas is mainly governed by the electrons having the same charge as the ions, and therefore feeling the same electric force, but are less inertia-controlled due to their smaller mass.

5.1 Ionization Degree and Coupling Constant At non-zero temperatures any gas contains, along with the neutrals, a certain number of ionized atoms and electrons. Their concentration is small if the temperature is considerably below the ionization energy. For a plasma with no refueling or recycling and no transport losses, the neutral density would rapidly drop to a low value, which is the equilibrium value obtained by balancing the recombination rate with the ionization rate at a certain electron temperature. The degree of ionization is defined as the ratio of the electron density to the total density of the electrons and neutral atoms ne nn + ne

(5.3)

not to be confused with the ionization ratio ne /nn . When the average energy of the particles approaches the ionization energy, the gas almost completely turns into an ionized plasma. In thermal equilibrium, the particle velocity distribution is a Maxwellian distribution (A.1). Often, a partial equilibrium is observed, where the temperatures determining the distributions for the electrons and the ions are different.

5.1 Ionization Degree and Coupling Constant

33

Even at very low concentrations of charged particles, their interactions dominates the behavior of an ionized gas. The cross-section for binary collisions of charged particles (see Sect. 5.4) σ=

e4 ln Λ 4 × 10−17 2 e4 ln Λ  = m 4π2o m2 v 4 36π2o (kB Te )2 Te [eV]2

(5.4)

is for Te ≤ 1 eV several orders of magnitude larger than the cross-section for ion-neutral collisions σn  10−20 m2 . The cross-section according to (5.4) becomes, however, very small at high temperatures and therefore close to σn . A plasma (with negligible losses) is almost fully ionized at electron temperatures of a few eVs, and the concentration of neutral atoms is therefore low. The main difference between the behavior of a plasma and a gas is due to the effects exerted by the electric and magnetic fields on the motion of the charged particles. The coupling constant Γcoupl , i.e., the ratio of the average Coulomb energy of particles separated by the mean distance to the thermal energy 1 Q2 (5.5) Γcoupl = 4πo d kB T characterizes a system of interacting charged particles, where Q is the charge of the particles. The average distance between the particles is defined by the density n
1/3 3 d= . (5.6) 4πn The boundary between ideal and non-ideal plasmas is given by Γcoupl = 1. When the ratio (5.5) exceeds a critical value of about 170 solidification occurs. In gaseous plasmas, this condition would require, for example, a density of
n>

170

4πo 31/3 kB T e2 (4π)1/3

3 = 4 × 1032 (T [eV])3

1 1 = 4 × 1029 3 3 m m

(5.7)

for a plasma temperature of 0.1 eV. In dusty plasmas, this condition can be satisfied. Small particles, grains embedded in a plasma are negatively charged by the incoming electrons, as every solid object does during plasma contact (see Chap. 7). The collected charge on a grain of only 0.1 µm radius can be very large (several thousand elementary charges) and the distance between the grains is small in a dense plasma. In this case, the electrostatic interaction energy between grains can become comparable to the thermal energy of their random motion, and the criterion Γcoupl > 170 can be met. Thus, the grains can form regular structures or lattices—so-called Coulomb or plasma crystals, as theoretically predicted in 1986 by Ikezi [64]. The critical value of the coupling parameter depends on plasmas of the shielding effect acting on charged particles embedded in a uniform background of neutralizing charge. It increases nearly exponentially with d/λD [65].

34

5 The Plasma State

Dusty plasmas as an ensemble of charged dust particles immersed in a plasma occur in planetary ring systems, interstellar clouds, and tails of comets [66, 67] but also on earth in plasma technology applications such as etching, plasma assisted deposition, or sputtering. The resulting contamination problems can only be reduced by a better understanding of the complex behavior of dusty plasmas with their tendency to self-organize into ordered structures. Analyzing (the rather large) plasma crystals by both simulations and experiments offers the chance for detailed studies of solidification in three dimension, melting, and phase transitions [68, 69].

5.2 Debye Length In the case of no magnetic field, the Debye length turns out to be the characteristic length describing the screening of a single test charge Q in a plasma by the surrounding electrons and ions. The density distributions of the electrons ne (r) and ions ni (r) as a function of distance r from this test charge are described by ne (r) = npl exp[eφ(r)/(kB Te )]

(5.8)

ni (r) = npl exp[−eφ(r)/(kB Ti )]

(5.9)

where npl is the undisturbed plasma density far away (npl = ne (r → ∞) = ni (r → ∞)), Ti and Te the temperature of ions and electrons, respectively, and φ(r) is the electric potential distribution. Using the Poission equation ∇ · E(r) = ρ(r)/o ,

E(r) = −∇φ(r)

(5.10)

where the charge density is given by ρ(r) = e[ni (r) − ne (r)] + Qδ(r)

(5.11)

we obtain enpl Q {exp[−eφ(r)/(kB Ti )] − exp[eφ(r)/(kB Te )]} = − δ(r) o o (5.12) with the Dirac function δ(r). With the condition that the kinetic energy is much higher than the potential disturbance caused by the test charge ∇2 φ(r) +

eφ(r) kB Te,i

(5.13)

(5.12) simplifies to ∇2 φ(r) − (1/λ2D )φ(r) = −

Q δ(r) o

(5.14)

5.3 Plasma Frequency

35



with λD =

o kB Te kB Ti . npl e2 (kB Te + kB Ti )

(5.15)

The Debye length does not depend on the mass of the involved particles, i.e., inertia does not play a role here, but is affected by the smaller temperature of both plasma constituents—the smaller the kinetic energy of one species, the smaller the region of potential disturbance as well. In spherical coordinates, the solution of (5.14) is φ(r) =

1 Q exp(−r/λD ) 4πo r

(5.16)

which is valid for r = 0 and satisfies the condition φ(r) → 0 for r → ∞, and φ(r) = Q/(4πo r) for r → 0. Accordingly, we obtain for the density distribution (exp(±eφ(r)/kB T ) ≈ 1 ± eφ(r)/kB T )
 1 1 2 ρ(r) = −npl e φ(r) + + Qδ(r) kB Ti kB Te Q =− exp(−r/λD ) + Qδ(r) . (5.17) 4πrλ2D The test charge Q causes a spatial rearrangement of the charged particles in a way that the disturbance is as small as possible. Of course, one species alone is able to realize the screening, the other might be fixed. Out of the sphere with radius λD , the effect of the test charge almost vanishes. The process of charge screening, however, should not be considered as a static one as might be suggested by the above analysis. The charged particles oscillate around a test charge driven by continuously rising and decaying electric fields and inertia. The distribution (5.17) is obtained only be averaging over several oscillation periods.

5.3 Plasma Frequency Having small density fluctuations in a fully ionized plasma ne (r, t) = npl + ne (r, t)

and ni (r, t) = npl + ni (r, t)

(5.18)

the continuity equations of both species can be linearized ∂ne,i + npl (∇ · v e,i ) = 0 ∂t

(5.19)

as well as the momentum equations assuming constant temperatures Qe,i ∂v e,i kB Te,i = E− ∇ne,i − νe,i v e,i ∂t me,i me,i npl

(5.20)

36

5 The Plasma State

with the charges Qi = qi e and Qe = −e. Neglecting collisions and the pressure term, since a small density disturbance is already sufficient to generate large electric fields, allows for the removal of the temperature dependence. For singly charged ions (qi = 1), the electric field is given by ∇·E =

ρ e e = (ni − ne ) = (ni − ne ) . o o o

(5.21)

Taking the divergence of (5.20) and combining (5.19) with (5.21), we obtain for each species ∂ 2 ne,i Qe,i npl Qe,i npl e  =− ∇·E =− (n − ne ) . 2 ∂t me,i me,i o i

(5.22)

Substituting npl = ni − ne yields ∂2  e2 npl  e2 npl  (ni − ne ) = − (ni − ne ) − (n − ne ) 2 ∂t mi o me o i

(5.23)

and thus

∂ 2 npl 2  + ωpl npl = 0 . (5.24) ∂t2 The plasma density disturbance npl varies harmonically according to the plasma frequency 
  1 npl e2 npl e2 (me + mi ) 1 + . (5.25) ωpl = = o me mi o me mi In the case of me mi , (5.25) reduces to the well-known electron plasma frequency  ωpe = npl e2 /(me o ) (5.26) often simply called the plasma frequency.

5.4 Collisions in Plasmas A particle does not perform a collision during a time ∆t with the probability fP (∆t) = exp(−∆t/τ )

(5.27)

where τ = 1/ν is the average time between two collisions and ν the collision frequency. This relation is readily obtained from the balance equation for particles that have not participated in collisions after a time t N (t) dN =− dt τ

→

N (t) = No exp(−t/τ )

(5.28)

where N is the number of particles and No = N (t = 0). The collision time τ , the mean free path λ, and the velocity of the particle v are simply related to

5.4 Collisions in Plasmas

37

each other, via λ = τ v. In other words, by crossing a plasma slab of thickness ∆x the incoming flux density Γin is reduced by collisions Γout = Γin − Γin n S ∆x

σ S

(5.29)

where n is the particle density in the plasma, S is the surface area of the plasma slab, and σ is the cross-section. The ratio σ/S is a measure of the collision probability, and the term n S ∆x gives simply the number of particles in the plasma slab. For the change of the flux density, we obtain Γout − Γin ∆Γ dΓ = = = −nσΓ ∆x ∆x dx

(5.30)

with the solution Γ = Γo exp(−x/λ) where λ = 1/(nσ) is the mean free path, τ = λ/v is the mean time between two collisions, and ν = nσv is the collision frequency. In partially-ionized plasmas, collisions with neutral atoms may be important. The corresponding cross-section is simply given by σ = πro2  10−20 m2

(5.31)

where ro is the gas-kinetic radius. In a fully ionized plasma, the motion of each charged particle is mainly governed by the long-range Coulomb interaction with the force F = e2 /(4πo r2 ), where r is the distance between both (singly charged) particles. A strong deflection (≥ 90o ) is always accompanied by a change of momentum, i.e., the forward directed momentum is lost, ∆v =F ∆t ∆(mv) mv e2 ≈ = . ∆t ∆t 4πo r2 m

(5.32)

The time of interaction ∆t can be estimated with ∆t = r/v yielding r=

e2 4πo mv 2

(5.33)

and the cross-section for large deflections is σ = πr2 =

e4 . 16π2o m2 v 4

(5.34)

In fact, r is equal to the impact parameter required for a 90o deflection and twice the value of distance ∆ at which the potential energy is equal to the kinetic energy of the incoming particle mv 2 e2 = 4πo ∆ 2

→

∆=

o e2 = 2r = 2ρ90 . p 2 2πo mv

(5.35)

38

5 The Plasma State

However, small-angle scattering contributes mainly to the cross section of binary collisions. Therefore, the effective cross-section is by a factor of 4 ln Λ larger than given by (5.34). The Coulomb logarithm ln Λ varies typically from 5 for cold and dense plasmas to about 20 in the case of hot fusion plasmas. The corresponding collision frequency is then (see (5.34)) ν = nσv =

ne4 ln Λ 4π2o m2 v 3

or

ν=

4π2o

ne4 ln Λ √ m (3kB T )3/2

(5.36)

(with mv 2 /2 = 3kB T /2). The symbol ν is used for electrons with mass me higher than ions with mass M by the factor  M νei =  40 . (5.37) νii me Usually, the collision frequencies are obtained by taking into account the velocity distributions of both the “test” and “field” particles (Sect. A.1) [14]. The exchange of momentum in collisions results in friction, when different particles are involved, and in viscosity in the case of collisions between particles of the same species. The friction force FR = (flux density) × (cross − section) × (momentum transfer) = (collision frequency) × (momentum transfer) = (nv) · σ · (mv) equals the change of momentum in time. Therefore we obtain  o 2 FR = 4 ln Λ nmπ ρ90 v2 p

(5.38)

(5.39)

o

where ρ90 is the impact parameter for a 90o deflection. In the equations of p plasma physics, this force appears, for example, as (in N/m3 ) j (5.40) e expressing the force acting on electrons due to their collisions with the plasma ions. Bearing in mind that each plasma species has its own velocity distribution, the term (v i − v e ) is the velocity difference of both species. Note that νei = νie but the friction forces—according to Newton’s law “actio” equals “reactio”—are the same, i.e., Fei = −Fie . Several physical effects in plasmas besides friction such as mobility, diffusion, and conductivity are closely connected to collision processes. From the force balance enE = ∇p + m n ν v (5.41) F R = ne me νei (v i − v e ) = me νei

we obtain for an isothermal plasma the solution v=

kB T ∇n ∇n en E− = µb E − D mnν mν n n

(5.42)

5.4 Collisions in Plasmas

39

with the mobility coefficient µb = e/(mν) and the diffusion coefficient D = kB T /(mν). This description simplifies the process of acceleration in the electric field between two single collisions and the slowing down process during collisions. In the case of no pressure gradient, the current density according to (5.42) j = env =

e2 n E = σc E mν

(5.43)

is characterized by the coefficient σc σc =

4π2o (3kB T )3/2 e2 n √ = mν e2 ln Λ m

or in general σc =

e2 n  m k νeV

(5.44)

called plasma conductivity, where the collision frequency is taken from (5.36). It depends strongly on plasma temperature T but not on plasma density, since both the collision frequency and the current density are proportional to the particle density. The exact value of the plasma conductivity including also the effect of the effective charge is given by [70]   3/2 −1/3 Te [eV ] 1 . σc = 1.92 × 104 2 − Zeff Zeff ln Λ Ωm

(5.45)

A clean H-plasma with Te = 100 eV has about the same resistivity as stainless steel (7 × 10−7 Ω m). At a temperature of about 1 keV, its conductivity comes close to the conductivity of copper (2 × 10−8 Ω m). With higher plasma temperatures, the conductivity increases up to even higher values, when a plasma current of several MA is driven by a loop voltage of only a few volts in tokamak experiments. Several relaxation times can be introduced. The “slowing down time” is defined by v (5.46) ts = − ∆v/∆t with the average velocity change per unit time ∆v/∆t. For small velocities, ts is near constant, and for higher velocities ts ∝ v 3 . The “deflection time” is given by v2 (5.47) td = (∆v⊥ )2 /∆t with the average change of the square of the velocity component perpendicular to the initial direction of the test particle (∆v⊥ )2 /∆t. The change of energy in a collision can be expressed as M 2 M = 2

∆E =

 1 2 (v + ∆v )2 + ∆v⊥ − M v2 2   2 ≈ M v∆v . 2v∆v + ∆v2 + ∆v⊥

(5.48)

40

5 The Plasma State

Hence, we have for the “energy exchange time” tE =

M 2 v4 v2 E2 = = (∆E)2  4M 2 v 2 (∆v )2 /∆t 4(∆v )2 /∆t

(5.49)

with the average change of the square of the velocity component parallel to the initial direction (∆va )2 /∆t. One important parameter is the “self-collision time” tc . It determines the time required to establish thermal equilibrium by collisions after a disturbance. For test and field particles of the same kind moving with the average thermal velocity, we have approximately [71] √ 2 M (kB T )3/2 2π2 M 2 v 3 ≈ o . (5.50) tc ≈ tE ≈ tD ≈ 4 o4 Z e n ln Λ Z 4 e4 n For a hydrogen plasma with a temperature of kB T = 1 keV and a density of n = 1019 m−3 , the self-collision time is about tc = 0.001 s. In the case of kB T = 100 eV, we obtain tc = 3 × 10−5 s. This characteristic time increases strongly with temperature, and it is much larger for ions than for the light electrons. With an average distance between the particles of about ∆l =

1 1/3 npl

= 1.15 × 10−7 m = 1150 ˚ A

(5.51)

for a plasma density of npl = 1020 1/m3 , it is useful to distinguish binary collisions, i.e., to neglect here the interaction of one particle with all particles o for a 90o in the near and far neighborhood, since the impact parameter ρ90 p deflection of an electron with an energy of E = 10 eV at a proton o

ρ90 = p

e2 (me + mp ) = 7.2 × 10−11 m = 0.72 ˚ A 8πo mp E

(5.52)

is small in comparison to ∆l . The exact derivation of (5.52) is given in Sect. 6.1 (see (6.34)). Even for a 1 degree deflection, the corresponding impact parao o −9 m = 82 ˚ A < ∆l . meter is rather small, i.e., ρ1p = ρ90 p cot(α/2) = 8.2 × 10 For larger impact parameters, ρp > ∆l , the model of binary collisions becomes questionable. Nevertheless, up to a value of ρp = λD (λD = o o kB Te /(npl e2 ) = 2 × 10−6 m for kB Te = 10 eV, i.e., λD  ∆l  ρ90 p ) multiple collision events can be thought as a sequence of several binary collisions. With this assumption, the collision cross-section is usually derived by integrating in the interval ρp = [0, λD ]. This integration leads to the additional factor 4 ln Λ mentioned above. In contrast to solids, the charged particles in plasmas are not fixed at rigid lattice positions. They react sensitively in electric and magnetic fields by rearrangement, charge separation, and induced currents. The influence of the self-generated fields back on the particle motion itself leads to the flexible and chaotic appearance of plasmas, which try to evade any manipulation.

5.5 Transport Processes in Plasmas

41

Each single particle feels the surrounding plasma particles and moves in their field. Such terms as mean free path or collision time are not fully adequate to deal with the true many-particle interactions in plasmas. Collective processes dominate on a spatial scale equal and larger than the Debye length. Introducing a density description (Sect. 5.7) allows the determination, via the Poisson equation, of the electric field based on a certain spatial distribution of charged particles. Inside the Debye sphere, the local electric fields have to be considered as well. However, these fields are small in the case of a symmetrical arrangement of the particles. The more particles are included in the Debye sphere, the less local fluctuations play a role. According to the described methodology collective effects, i.e., manyparticle collisions, are more or less artificially separated from binary collisions even though the physics behind them, i.e., the Coulomb interaction, is the same. A plasma is often called collisionless when the absence of only the binary collisions with impact parameters smaller than the Debye length is meant.

5.5 Transport Processes in Plasmas Transport effects arise whenever inhomogeneities, for example in density or energy, occur in plasmas. Several parameters help to classify transport phenomena. The degree of ionization decides which kind of collisions are important. The direction of the magnetic field is also the direction of maximum transport, since every process of momentum and energy transfer is much more efficient along the magnetic field lines than across them. The ratio of the thermal energy density of electrons and ions to the magnetic pressure determines the degree of confinement, which is limited due to arising instabilities at larger ratios. The ratio of the ion gyro-radius to the characteristic length scale of the gradients of the plasma parameter gives the regions of validity of the different simplifying theoretical models of plasma transport [14, 35, 36]. Any real plasma will have a density gradient. The plasma particles diffuse toward regions of low density, thereby tending to flatten out the density gradient. If the plasma is only weakly ionized, then the charged particles collide primarily with neutral atoms rather than with one another. In ionized plasmas collisions of charged particles dominate. Like-particle collisions, e.g., ion–ion collisions, do not lead to cross-field diffusion, since the gyration centers of two particles being involved in a collision move exactly equal but opposite distances across the magnetic field. For each particle that moves outward, there is another that moves inward. The transport due to unlike-particle collisions, e.g., electron–ion collisions, is ambipolar, since the displacements of the gyrations centers of both particles are equal for each collision [36]. The diffusion process can be described introducing the concept of a “random walk”. In the direction of the magnetic field, the more or less free transport of the charged particles is reduced by collisions. The transport

42

5 The Plasma State

perpendicular to the magnetic field lines is, on the other hand, enhanced by collisions. During collisions the particles change their positions in a random walk across the field. The step length in this random-walk process is about the gyro-radius, which is usually much smaller than the mean free path between two collisions that characterized the random motion along the magnetic field lines. Global drifts of charged particles (Sect. A.1) usually do not lead to radial transport, rather to rotation of a plasma column. The net fluxes of electrons and ions are adjusted to retain the high degree of charge neutrality as required in plasmas. This process of adjustment involves the generation of electric fields as soon as a slight charge imbalance occurs. Diffusion involves the transport of momentum in the plasma. The flux of particles to the density gradient is characterized by the diffusion coefficient. Thermal conduction is connected to the transport of kinetic energy. The thermal conductivity relates the heat flux to the temperature gradient. Besides binary collisions, collective effects dominate the transport in magnetized plasmas. Local drifts caused by microinstabilities enhance the transport across the magnetic field by several orders of magnitude in comparison with the transport effects due to binary collisions. Analyzing experimental data of different fusion experiments, Fussmann concludes that the anomalous large transport of impurities does not depend on the impurity mass, charge, or velocity [72]. This serves as a clear indication that electric drift effects are responsible for the enhanced transport. The origins of anomalous diffusion are assumed to lie in fluctuating electric fields, which exceed the thermal level. 5.5.1 Transport by Binary Collisions In the classical picture, the gyrating charged particles are fixed to the magnetic field lines. Radial displacements are only possible by collisions. The corresponding diffusion coefficient for cross-field transport is, in the frame of the random-walk model, given by the collision frequency ν and the square of the mean step length, which is about the gyro-radius. For electrons, we have then (me ve2 /2 = 3kB Te /2)
2 ve 3 me kB Te classical 2  ν ρce = ν ν . (5.53) D⊥ ωce e2 B 2 The prefactor 1/(2d) in (6.109) has been omitted for simplification. Only unlike-particle collisions lead to transport, and we have with (5.36) e4 n ln Λ 3 me kB Te 1 √ 2 3/2 4πo me (3kB Te ) e2 B 2 √ e2 n ln Λ me √ = 4π2o B 2 3kB Te 2 n [1/m3 ] m  3 × 10−22 . s (B [T])2 Te [eV]

classical = νei ρ2e = D⊥

(5.54)

5.5 Transport Processes in Plasmas

43

For example, taking B = 2 T, Te = 10 eV and n = 1019 1/m3 , (5.54) yields a classical = 2.4 × 10−4 m2 /s. Important to very small diffusion coefficient of D⊥ note is that the same result is obtained when considering the plasma ions (if Ti = Te ), since (5.55) νei ρ2e = νie ρ2i due to νie = (me /Mi ) νie . In the fluid picture, the simplified Ohm’s law [35] E + v × B = η j  + η⊥ j ⊥

(5.56)

is used together with the pressure balance j × B = ∇p

(5.57)

where j is the current density, η the resistivity, and p the plasma pressure p = (kB Te + kB Ti )n. Taking the cross product of (5.56) with B and replacing j × B from (5.57) yields the fluid flow velocity across the magnetic field v ⊥ = −η⊥

∇p E × B + B2 B2

(5.58)

where the first term describes the transport caused by collisions, and the second term is the electric drift term. For the collision driven flux density, we obtain η⊥ (kB Te + kB Ti )∇n classical = −D⊥ ∇n (5.59) nv⊥ = −n B2 the familiar result (5.54), since the resistivity can be represented as η = me νei /(e2 n) ∝ 1/(kB Te )3/2 . 5.5.2 Neoclassical Diffusion In toroidal devices such as tokamaks and stellarators, the magnetic field has to be twisted in order to eliminate gradient-B and curvature drifts. While a charged particle moves along a magnetic field line, it comes into regions with higher magnetic field strength near the inner wall of the torus and regions with weaker magnetic field strength near the outer wall at larger radii. A magnetic mirror configuration is therefore established and some particles are trapped. They do not circulate around the torus, but follow trajectories whose projections on the torus cross-section form banana-shaped orbits. By performing collisions, the particles are able to hop from one banana orbit to another. The resulting diffusion across the magnetic field is then characterized by the width of the banana orbit, which is usually much larger than the gyro-radius. This effect, called neoclassical diffusion, enhances the transport by a factor of 10 to 100 in comparison to classical diffusion. These values are, nevertheless, still too small to explain the large transport observed in fusion experiments. In addition, the neoclassical theory leads to some predictions and dependencies which could not be confirmed in experiments [73–75].

44

5 The Plasma State

5.5.3 Anomalous Transport The expected strong dependence of transport on the magnetic field strength B according to the classical theory, i.e., D⊥ ∝ 1/B 2 , in the beginning of fusion research gave rise to hope of quick realization of controlled fusion. Soon, an anomalously large diffusion, which exceeded the calculated one by four experiment  orders of magnitude, was observed in the first experiments, i.e., D⊥ 3 4 classical . Furthermore, the experimental diffusion coefficient DB = (10 –10 ) D⊥ scales were more like 1/B. Developing an arc discharge with magnetic field for uranium isotope separation, Bohm gave (without derivation) the following semi-empirical formula to describe this anomalous diffusion [76] D⊥ =

1 kB Te ≡ DB . 16 e B

This relation may be expressed using the random-walk approach as  √ kB Te kB Te kB Te me = DB  ve ρce = ωce λD ρce = me eB eB

(5.60)

(5.61)

where ωce is the electron gyro-frequency and λD is the Debye length. Equation (5.61) does not provide, √ however, a reasonable, physical explanation of (5.60), since a step length of λD ρce is of no physical relevance. The scaling of DB with kB Te and B can be shown to be the natural one, whenever the losses are caused by E × B drifts due to fluctuating electric fields E [77]. The local particle flux density is then proportional to the drift velocity v⊥ E (5.62) Γ⊥ = nv⊥ ∝ n . B Because of Debye shielding, the maximum potential in the plasma is given by eφmax ≈ kB Te . If ∆ is a characteristic scale length of the plasma, the maximum electric field is then Emax ≈

kB Te e∆

(5.63)

leading to kB Te n kB Te ≈− ∇n = −DB ∇n . (5.64) ∆ eB eB With the prefactor 1/16, the relation (5.60) agrees surprisingly well with various experiments. Well applicable to the diffusion in the rather cold edge plasma of fusion experiments with DB  1 m2 /s, it is clearly not valid for the hot core region, since there it would suggest much too high values of the diffusion coefficient. Although the anomalous transport is often treated in the frame of a diffusion description, it is actually governed by convective terms (“zonal Γ⊥ ≈

5.6 The Vlasov Equation

45

flows”) [78, 79] and the behavior of vortex structures (“blobs”) in magnetized plasmas [80]. These turbulent eddies have a radial extent of 1– 2 cm and a typical lifetime of about 0.5–1 ms. By poloidal shear flows, i.e., vpoloidal = E radial × B toroidal , the eddies can be radially decorrelated resulting in suppression of cross-field transport. The Kelvin-Helmholtz instability as well as the Rayleigh convection instability occur not only in fluids but also in magnetized plasmas due to non-uniformity of plasma flow (shear flow) and non-uniformity of temperature, respectively. Despite the effort and progress made in the field of plasma transport theories [81–87], reliable predictions can hardly be given up to now. In estimations and computer simulations the transport coefficients are usually given as constant input parameters, for example D⊥ = 0.1–1 m2 /s. The exact solution of the Vlasov equation (see below) by numerical methods or by using a PIC simulation opens, in principle, the opportunity to analyze the “anomalous” transport, since all the necessary physics is included. Such a task, however, would clearly overload existing computer systems.

5.6 The Vlasov Equation The use of the Boltzmann kinetic equation represents one of the most comprehensive approaches to describe the collective dynamics in many-particle systems such as plasmas. In a very useful approximation, the Vlasov equation ∂f ∂f ∂f df = +v· +a· =0 dt ∂t ∂r ∂v

(5.65)

the collision term on the right-hand side of the Boltzmann equation is omitted, but the internal electric and magnetic fields are included in the force term a=

F ext Q F ext + F int = + (E int + [v × B int ]) . m m m

(5.66)

The Vlasov equation describes the evolution of the particle distribution function f = f (r, v, t) in time, where the force F is given by externally applied fields and by internal electromagnetic fields, i.e., E int and B int . In many applications, the magnetic field is almost constant and given by external sources. In these cases, the plasma currents are too small to generate considerable internal magnetic fields, and the plasma pressure is much smaller than the magnetic pressure B 2 /(2µo ). However, the internal electric field E = −∇φ cannot be neglected. It should be obtained using the Poisson equation ∇2 φ = −

ρ o

where the electric potential φ is determined by the charge density ρ  ρ(r) = Q f (r, v, t) dv .

(5.67)

(5.68)

46

5 The Plasma State

Analytical solutions of the self-consistent set of equations (5.65–5.68) are available for only very simplified situations. In most of the cases of interest, extensive numerical calculations are required. In principle, all collision processes—including even binary collisions—can be described within that approach. This is at first glance somewhat surprising, since the collision term of the Boltzmann equation has been disregarded. However, the internal fields are obtained by a macroscopic smoothing of the local fields between the colliding particles. It is rather a question of spatial resolution, which is applied in the process of density calculation knowing the position of particles (Sect. 5.7), than a principal one. The characteristic length usually applied in the smoothing procedure is the Debye length, so one can argue that in this case collisions with impact parameters smaller than λD are neglected. More importantly, the collective processes, i.e., the many-particle interactions, which dominate the plasma behavior are still well-treated. The evolution of the particle distribution function can be described by following the motion of the particles through phase space. The set of equations (5.65–5.68) can be solved, therefore, by evaluating the trajectories m

dv = Q (E + [v × B]) dt

(5.69)

of all particles, since the points on the enclosing surface of any volume in phase space move according to dr/dt = v. For the next time step, we have


v
= v +

t




Q (E + [v × B]) dt m

and

t

r = r +

t

v dt .

(5.70)

t

The particle density is obtained directly from the positions of the particles and not via the distribution function according to (5.68). Knowing the positions and velocities of all particles for all moments of time, the distributions function is well-defined. Such a numerical simulation (see Sect. 7.6) is fully equivalent to the self-consistent solution of the Vlasov equation together with the Poisson equation.

5.7 The Poisson Equation By defining a certain interaction law, for example the Coulomb interaction law, the force acting on one particle is given by the spatial positions of all other particles with respect to the position of the particle under consideration. A point-like charge qi e generates the electric field E i (r) =

qi e R i 4πo Ri3

(5.71)

5.7 The Poisson Equation

47

at a position given by the radius vector r, where Ri = r − r i and r i is the radius vector of the test charge qi e. The flux of the electric field E i through a closed surface is  (5.72) ΓE,i = (n · E i )dS S

where n is the normal vector of the surface element dS. From position r i the surface element is seen under the solid angle element (Fig. 5.1) dΩ = dS(n · Ri )/(Ri )3 .

(5.73)

Considering (5.71), we obtain from (5.72)   qi e qi e 4π 4π for r i ∈ V o ΓE,i = dΩ = 0 for r i ∈ V 4πo

(5.74)

S

where the volume V is enclosed by the surface S. The field generated by several charges   qi e R i E(r) = Ei = (5.75) 4πo Ri3 i i is the superposition of the various individual fields. The general flux of the electric field yields   1  Qt ΓE = (n · E)dS = ΓE,i = qi e = (5.76) o o i i∈V

S

dS Æ

n

dW i Æ

Ri

V Qi

Æ

r

z

Æ

ri y

S x Fig. 5.1. Considered geometry in the derivation of the Poisson equation

48

5 The Plasma State

where Qt is the total charge enclosed in the volume V . Dividing the space now into small volume elements ∆Vk with their centers r k , a charge density can be defined 1  ρ(r) = qi e (5.77) ∆Vk i∈∆Vk

leading in the limit ∆Vk → 0

 Qt =

ρ dV .

(5.78)

V

Under the assumption of Ri = r − r i ≈ r − r k = Rk , (5.75) is replaced by E(r) =

  k i∈∆Vk

 R k  qi e qi e R i ≈ 4πo Ri3 Rk3 4πo k

 Rk ρ(r k ) = ∆Vk Rk3 4πo

i∈∆Vk

(5.79)

k

where the summation can be transformed into an integration  ρ(r  ) R dV  E(r) = 4πo R3

(5.80)

with R = r − r  . A similar presentation can be given for the electric potential  ρ(r  ) dV  1 . (5.81) φ(r) = 4πo |r − r
| V

Taking (5.76) and (5.78) together with the Gauss law   (n · E)dS = divE dV S

(5.82)

V

yields the Poisson equation divE =

ρ o

and ∇2 φ = −

ρ o

(5.83)

since E = −∇φ. Of course, (5.81) is a solution of (5.83). To prove that, it is inserted into the Poisson equation
   1 1 1 ρ(r  )dV   2 = ∇2 ρ(r )∇ (5.84) dV  . 4πo |r − r
| 4πo |r − r
| V

V

Using the relation


∇

2

1 |r − r  |



= −4πδ(r − r  )

(5.85)

5.7 The Poisson Equation

49

this leads again to (5.83) 1 ∇ φ= 4πo



2

ρ(r  )[−4πδ(r − r  )]dV  = −ρ(r)/o .

(5.86)

V

Without losing universality, the interaction between particles can be described in term of fields instead in terms of forces. The transition from point-like particles to a density description marks the transition from the Coulomb force to the Poisson equation (5.67) that relates the distribution of charged particles to the electric field. One particle can be thought of as being represented by a localized “charge cloud”. The calculations via fields is as accurate as the spatial resolution. Just inside the charge cloud and in its direct neighborhood inaccuracies do occur. The interaction between the charge clouds is taken exactly into account at larger distances. The numerical solution of the Poisson equation based on efficient algorithms is much faster than the calculation of the interaction between all particles (Sect. 7.6.1). In addition, the number of representative particles, the charge clouds, used in the simulation is usually several orders of magnitude smaller than the number of particles in real plasmas.

6 Particle Coupling

The interaction of plasma particles with solid surfaces can be classified according to the particles under consideration: (1) bombardment by atomic particles, in particular neutral atoms formed by charge exchange, molecules, and ions, (2) electron bombardment, (3) interaction with electromagnetic radiation, and (4) neutron irradiation. While neutrons cause a lot of damage in the bulk material (Sect. 12.5), the sputtering coefficient is smaller than 10−4 , and this effect can be neglected. The effect of electrons [88] and photons on irradiated surfaces reduces primarily to heating. The resulting temperature gradient creates thermal stresses, which impose a limit on the maximum thickness of the material through which the heat flux is removed (see Sect. 8.3). Due to the impact of the particles, a number of interaction and transformation processes are induced in the material leading to a change of its structure and composition, and to emission of particles. A number of processes have to be considered: 1. 2. 3. 4. 5.

6. 7. 8. 9. 10.

Erosion processes (sputtering, evaporation, etc.) (Sect. 6.5) Backscattering of ions and electrons from the surface (Sect. 6.8) Emission of electrons (Sect. 6.9) Ion implantation and release of gaseous deuterium, tritium, and helium (Sect. 6.4) Diffusion of gas atoms and defects, the formation and decay of mobile and immobile complexes of implanted atoms and defects, and the solubility of the gas in the material (Sect. 6.3) Segregation (Sect. 6.3) Desorption of impurities during ion and electron impact Blistering (Sect. 6.5.5) Changes in a surface layer subjected to ion irradiation, changes in the chemical and phase composition (Sect. 6.3) Chemical processes at the surface

52

6 Particle Coupling

11. Inelastic processes during the interaction of ions with the surface (changes in the charge state [89], ion, electron and photon emission) (Sect. 6.6) Each particle that strikes a material surface is subjected to collisions (predominantly elastic) with lattice atoms and predominantly inelastic collisions with the electrons of the bulk material [90–93] (Sect. 6.2). During these interactions, the particles change their initial direction of motion and their charge state, lose their kinetic energy (Sects. 6.2.2 and 6.2.3) and momentum. They come to rest by reaching energies in the eV range, i.e., less than the binding energies. At these low energies, the particle is in energetic equilibrium with the target. This process is called implantation (Sect. 6.3). The momentum transfer from the impinging particles to the lattice atoms can lead to developing cascades of knock-on atoms, which may also leave the surface by surmounting the surface potential determined by the sublimation energy. These sputtered particles contribute to erosion and plasma contamination. Surface erosion (see Sect. 6.5) primarily determines the lifetime of components and the source of impurities, which increase the plasma radiation. Almost all particles emitted by sputtering leave the surface as neutrals. The basic model considering transport of particles in matter is the assumption of binary collisions (Sect. 6.1). As discussed in the case of collisions in plasmas (Sect. 5.4), this assumption holds roughly—in the case of solids—as long as the impact parameter for large deflections is smaller than the interatomic distance of about 2–3 ˚ A. Putting in an impact parameter equal to that o ˚ =2–3 A , the energy of particles should be larger than distance, i.e., ρ90 p Eo >

Z1 Z2 e2 (M1 + M2 )  Z1 Z2 · 3 eV o 8πo M2 ρ90 p

(6.1)

when additionally M1 /M2 1 is assumed. Another limit is the binding energy in the crystal lattice (several eVs), since the assumption of two-particle collisions is valid for free particles. Resonance scattering as well as nuclear reactions have to be considered separately and, of course, do not belong to the class of elastic collisions.

6.1 Binary Collisions For a particle of mass M1 with velocity vo that collides with a particle of mass M2 at rest (Fig. 6.1), the laws of energy and momentum conservation state M1 vo2 M1 v12 M2 v22 = + 2 2 2 M1 vo = M1 v1 cos θ1 + M2 v2 cos θ2 0 = M1 v1 sin θ1 − M2 v2 sin θ2

(6.2) (6.3) (6.4)

6.1 Binary Collisions M1

53

v 1 , E1

θ1

M1 M2

v0

θ2 v 2 , E2

M2

Fig. 6.1. Binary collision in the laboratory system (L-system)

taking into account that the trajectories of both particles during the collision lie in one plane (Fig. 6.1). Relations for sin θ2 and v2 can be deduced from (6.3) and (6.4), respectively, and inserted into (6.2) yielding
2

2  M1 2 2 v1 + vo2 − 2v1 vo cos θ1 (6.5) vo = v 1 + M2 ⎡

and v1 = vo

M1 ⎣cos θ1 ± M1 + M2




M2 M1

2

⎤ − sin2 θ1 ⎦ .

(6.6)

We have further sin θ2 sin(θ1 + θ2 ) 2M1 v2 = v o cos θ2 M1 + M2

v1 = vo

(6.7)

and sin2 θ2 sin2 (θ1 + θ2 ) 4M1 M2 E2 = E o cos2 θ2 (M1 + M2 )2 E1 = Eo

(6.8)

where E1 = M1 v12 /2 and E2 = M2 v22 /2. The obtained energy of the particle that was initially at rest is equal to the energy loss suffered by the incoming particle, i.e., E2 = ∆E. The introduction of the center-of-mass system (CM-system) facilitates the analysis of collision processes. In this coordinate system, the total momentum is zero, by definition. We have M1 uo1 + M2 uo2 = 0 and M1 u1 + M2 u2 = 0

(6.9)

where uo1 , uo2 are the velocities in the CM-system before collision; u1 , u2 are the velocities after collision. The center of gravity is at rest in the CM- system.

54

6 Particle Coupling n

u1 α

uo1

no

uo2 u2

Fig. 6.2. Binary collision in the center-of-mass system (no is the direction vector before and n the direction vector after collision)

Introducing the velocity of the CM-system v c with respect to the laboratory system (L-system), the velocities in both coordinate systems are simply related to each other uo1 = v o1 − v c = v o − v c ,

uo2 = v o2 − v c = −v c

(6.10)

where v o2 = 0 and v o1 = v o . After the collision, we have u1 = v 1 − v c and u2 = v 2 − v c (see Fig. 6.2). Using (6.10) in (6.9) gives the velocity of the center of mass v c = (M1 v o1 + M2 v o2 )/(M1 + M2 ) = M1 v o /(M1 + M2 ) .

(6.11)

With this relation together with (6.10), the velocities of the particle before the collision can be expressed in the CM-system by uo1 =

M2 v o , M1 + M2

uo2 = −

M1 v o . M1 + M2

(6.12)

The velocity vectors are turned only by an angle α during the collision (Fig. 6.2) u1 = uo1 n =

M 2 vo n , M1 + M2

u2 = uo2 n = −

M 1 vo n . M1 + M2

(6.13)

Their absolute values remain the same. Note that uo1 and uo2 as well as u1 and u2 are in opposition to each other. The transition back to the L-system is simply done by adding the center-of-mass velocity (Fig. 6.3) v 1 = u1 + v c ,

v 2 = u2 + v c .

(6.14)

As illustrated in the figure, the following relation holds that u1 sin α = v1 sin θ1 ,

u1 cos α + vc = v1 cos θ1

(6.15)

6.1 Binary Collisions

55

n vs u1 v1 α θ1 no

θ2

u2

v2 vs

Fig. 6.3. Velocities in the L-system and CM-system after collision

leading to tan θ1 = sin α/(cos α + vc /u1 ). Since vs /u1 = M1 /M2 as shown above, we obtain sin α tan θ1 = . (6.16) cos α + M1 /M2 In a similar way, the scattering angle of the second particle can be deduced θ2 =

π−α . 2

(6.17)

6.1.1 Scattering Angle α The conservation laws are not sufficient to determine the scattering angle. Its value must be calculated by assuming a certain interaction potential U (r) (not to be confused with the electric potential, since the interaction potential is given in energy units). The Lagrange function of the considered two-body system is M2 r˙ 22 M1 r˙ 21 + − U (|r 1 − r 2 |) (6.18) L= 2 2 with the relative radius vector r = r 1 − r 2 . If the center of the force field is in the position of the center of gravity, i.e., M1 r 1 + M2 r 2 = 0, then M1 r M1 + M2

(6.19)

r˙ 2 M1 M2 r˙ 2 − U (r) = µ − U (r) M1 + M2 2 2

(6.20)

r1 =

M2 r, M1 + M2

r2 = −

and the Lagrange function L=

describes the dynamics of one particle, which moves in a field with a fixed center. Thus, the two-body problem is reduced to the motion of one particle

56

6 Particle Coupling

with the reduced mass µ = M1 M2 /(M1 + M2 ) in a central field. The corresponding conservation laws of energy and angular momentum Mϕ are given in the polar coordinate system by  µ 2 r˙ + r2 ϕ˙ 2 + U (r) = const. (6.21) Er = 2 Mϕ = µr2 ϕ˙ = const. (6.22) where Er is the relative energy of the system. Replacing ϕ˙ in (6.21) as given by (6.22) yields  Mϕ2 2 dr = [Er − U (r)] − 2 2 (6.23) r˙ ≡ dt µ µ r or, after separation of r and t,  dr  t= 2 µ [Er − U (r)] −

2 Mϕ µ2 r 2

+ const.

(6.24)

If (6.22) is written in the form dϕ =

Mϕ dt µr2

(6.25)

then dt can be substituted by (6.23), and after performing an integration, we obtain the trajectory equation  (Mϕ /r2 )dr  + const. (6.26) ϕ= 2µ[Er − U (r)] − (Mϕ2 /r2 ) The scattering angle ϕ varies monotonically in time and does not change its sign. In the case of Mϕ2 U (rmin ) + = Er , (6.27) 2 2µrmin the radial velocity component is zero and the minimum distance rmin to the center of the force field is reached. The particle does not stop but turns back (Fig. 6.4). Due to the symmetric trajectory, the scattering angle α can be expressed as (6.28) α = |π − 2ϕo | . Instead of the constants Er and Mϕ , it is more convenient to apply the initial velocity vo of the incoming particle and the impact parameter ρp . We have Er = µvo2 /2 = Eo M2 /(M1 + M2 ) ,

Mϕ = µρp vo .

(6.29)

According to (6.26), the angle ϕo is given by the definite integral ∞ 

ϕo = rmin

with the limits of rmin and infinity.

1−

ρp r 2 dr ρ2p 2U (r) r 2 − µvo2

(6.30)

6.1 Binary Collisions

ϕo

ϕo

r

ρp

ϕ

57

α rmin

Fig. 6.4. Trajectory in the CM-system

6.1.2 Scattering in the Coulomb Field, U (r) = C/r For an interaction potential of the kind U (r) ∝ 1/r, the integration in (6.30) can be executed analytically, and we obtain for the scattering√angle in the CM√  system ( dx/(x x2 − a2 ) = (1/a) arccos(a/x), cos x = 1/( 1 + tan2 x)) ∞ 

α=π−2 rmin

r

ρp dr r2 − ρ2p − C r/Er


= 2 arctan

C 2ρp Er

 (6.31)

with Er = µvo2 /2 = M2 Eo /(M1 + M2 ) and C = Z1 Z2 e2 /(4πo ), where Z1 and Z2 are the atomic numbers of the particles. Another expression often used is α 1 = (6.32) sin2 2 1 + (2ρp Er /C)2 o

using tan2 x = sin2 x/(1 − sin2 x). The impact parameter ρ90 corresponding p to a scattering angle of 90 degrees is readily obtained, since for α = π/2 = 2 arctan(1) the condition C =1 (6.33) o 2ρ90 p Er must be fulfilled; thus, o

= ρ90 p

Z1 Z2 e2 (M1 + M2 ) . 8πo M2 Eo

(6.34)

6.1.3 Cross-Section The scattering of many particles moving with the same velocity is described by introducing the ratio dN dσ = [m2 ] (6.35) Γ

58

6 Particle Coupling

called cross-section, where Γ is the particle flux density. It relates the number of particles dN scattered by an angle in the interval [α, α + dα] per unit time to the number of particles crossing a unit area of the (homogeneous) beam during the same time interval. In the simple geometrical interpretation, the cross-section, more exactly, the total cross-section, defines the extension of an “active” area. Only by hitting this area, can a certain reaction occur. The number of particles with impact parameter ρp out of the interval [ρp , ρp −dρp ] is equal to the number of particles scattered by an angle α lying in the interval [α, α + dα] (6.36) dN = 2πρp Γ dρp . Using this in (6.35) leads to dσ = 2πρp dρp

or

   dρp (α)   dα . dσ = 2πρp (α)  dα 

(6.37)

In order to avoid negative cross-sections, the modulus is applied, since the derivative dρp /dα has, in most cases, gives a negative value. More frequently, the cross-section is given as a function of the solid angle dΩ = 2π sin α dα   ρp (α)  dρp (α)  dσ dΩ . (6.38) = dΩ sin α  dα  The transition from the CM-system to the L-system is trivial, if M2  M1 , since then, according to (6.16), θ1 ≈ α, µ ≈ M1 , and therefore dσCM = dσL . In general

  dσ dσ dΩL1,2 = dΩCM (6.39) dΩ L1,2 dΩ CM with dΩS = sin α dαdϕ, dΩL1 = sin θ1 dθ1 dϕ, and dΩL2 = sin θ2 dθ2 dϕ. Since the collision takes place on one plane, the azimuthal angle ϕ is the same. Applying (6.16) and (6.17), we have 1 cos α + M1 /M2 cos θ1 = ± √ = 2 1 + 2(M1 /M2 ) cos α + (M1 /M2 )2 1 + tan α cos θ2 = cos(π/2 − α/2) = sin(α/2) (6.40) and after some transformations dσ(α, ϕ) (1 + 2(M1 /M2 ) cos α + (M1 /M2 )2 )3/2 dσ(θ1 , ϕ) = dΩL1 dΩS |1 + (M1 /M2 ) cos α| dσ(θ2 , ϕ) dσ(α, ϕ) α = 4 sin . dΩL2 dΩS 2

(6.41)

Substituting for ρp from (6.31) together with its derivative ρp =

α C cot 2Er 2

and

dρp C 1 =− dα 4Er sin2

α 2

(6.42)

6.1 Binary Collisions

59

in (6.38), we find the cross-section for the Coulomb interaction (using cot (x) = −1/(sin x)2 , cot(x/2) = sin x/(1 − cos x) and 1 − cos x = 2 sin2 (x/2))

2 2 C Z1 Z2 e2 dσ 1 1 = = . (6.43) 4 α dΩ 4Er 8πo Er (1 − cos α)2 sin 2 This is the Rutherford cross-section. Summarizing, a detector which is positioned at angles (θ, ϕ) with respect to the beam axis and which covers a certain solid angle interval dΩ (given by the effective size of the detector divided by the square of the distance between the target and the detector) counts

part. dσ (θ, ϕ) dΩ dN = Γ Nt (6.44) dΩ s events per second, where Nt is the number of scattering centers in the beam– target interaction volume. Usually, thin foils are applied in the experiments to reduce the effect of multiple scattering. 6.1.4 Interaction Potential U (r) The collision process is mainly determined by the interaction potential, which must be chosen according to the situation under consideration. In plasma physics, usually, the Coulomb potential is applied and set to zero at distances larger than the Debye length (Sect. 5.4). Whether the calculations are performed with a “truncated” Coulomb potential or using the Debye length in an additional screening factor does not make much difference. The right choice of the interaction potential in solid state physics is more challenging since the screening effect of the electrons becomes important. Knowing the wave functions of all electrons would, in principle, allow for the determination of the exact interaction potential. However, the solution of the Schr¨ odinger equation is quite complicated even for simple atoms. Hartree suggested in 1927 a simplified method, which has been developed further by Fock and Slater [94–96]. Based on this approach, a number of potentials with different screening factors have been proposed in the past. The used screening lengths have the general form
2 1/3 9π −1/3 Z12 (6.45) as = ao 128 with the Bohr radius ao = 0.529 × 10−10 m and effective atomic number Z12 = (Z1x + Z2x )y

(6.46)

where x and y are chosen. In other approximations, the dependence on the distance is changed, for example U (r) ∝ 1/r2 . Such a functional dependence can only be valid in a certain, limited region away from the target atom. Thus, the average distance of approach should fall into this region. Such an

60

6 Particle Coupling

adjustment has to be made for each ion–atom pair of interest as a function of the ion energy. A list of potentials more frequently used is given below. 1. Potential of “hard spheres” U (r) =



∞ 0

for r < r1 + r2 for r > r1 + r2

(6.47)

where r1 and r2 are the radii of the colliding spheres. The corresponding cross-section is dσ = (r1 + r2 )2 /4 . (6.48) dΩ 2. Coulomb potential Z1 Z2 e2 (6.49) U (r) = 4πo r with the atomic numbers Z1 and Z2 . The cross section is (see (6.43))  a 2  a 2 dσ 1 1 s s = = (6.50) 4 α dΩS 4ε 2 ε (1 − cos α)2 sin 2 where the screening length as and the “reduced” energy ε 
 Z1 Z2 e2 as Eo M2 4πo = Er ε= 2 e Z1 Z2 (M1 + M2 ) 4πo as

(6.51)

have been introduced. It is worth noting that at higher energies and therefore closer distances of approach the Coulomb interaction potential becomes more and more realistic, since the incoming (high-energy) particle penetrates the electron shells deeply and see the “naked” atom core. The screening effect of the electrons is therefore reduced. 3. Firsov potential 0.415Z1 Z2 e2 aL 4πo r2 where aL is the Lindhard screening length  −1/2 2/3 2/3 aL = 0.8853 ao Z1 + Z2 U (r) =

(6.52)

(6.53)

with the Bohr radius ao . This potential finds its application in the energy interval 0.1 ≤ εL ≤ 1 with εL = aL Eo M2 4πo /(e2 Z1 Z2 (M1 + M2 )) (Lindhard reduced energy). The corresponding cross-section is dσ 0.415π 2 Z1 Z2 e2 aL (M1 + M2 ) π−α = dΩ 4πo Eo M2 α2 (2π − α)2 sin α 0.415 a2L π 2 (π − α) . = εL α2 (2π − α)2 sin α

(6.54)

6.1 Binary Collisions

61

4. Screened Coulomb potentials Z1 Z2 e2 Φ U (r) = 4πo r




r as

 (6.55)

where the function Φ(r/a) is represented as
Φ

r as

 =


 r ci exp −di , as i=1

n 

n 

ci = Φ(0) = 1 .

(6.56)

i=1

To this group belong the Bohr potential Φ(r) = exp(−r/as )

(6.57)

 −1/2 2/3 2/3 , the Moliere potential with as = ao Z1 + Z2


 r r r Φ(r) = 0.35 exp −0.3 +0.55 exp −1.2 +0.1 exp −6.0 (6.58) as as as with as = 0.8853ao

√ √ −2/3 Z1 + Z2 , the Kr-C potential

Φ(r) = 0.190945 exp(−0.278544 r/as ) + 0.473674 exp(−0.637174 r/as ) + 0.335381 exp(−1.919249 r/as ) (6.59) with as = 0.8853ao

√ √ −2/3 Z1 + Z2 , and the ZBL potential

Φ(r) = 0.028171 exp(−0.20162 r/as ) + 0.28022 exp(−0.40290 r/as ) + 0.50986 exp(−0.94229 r/as ) + 0.18175 exp(−3.1998 r/as )

(6.60)

with as = 0.8853ao /(z10.23 + Z20.23 ). Unfortunately, the cross-section as well as the relation of the scattering angle as a function of the impact parameter cannot be given in terms of elementary functions for these interaction potentials. 5. Morse potential U (r) = UD exp[−2C(r − ro )] − 2UD exp[−C(r − ro )]

(6.61)

where UD is the energy minimum at distance ro . The third parameter C can be used as well as the other two for fitting this analytical relation to a more realistic potential, for example, obtained by numerical calculations.

62

6 Particle Coupling

6.1.5 Binary Collision: General Case In the general case of binary collisions, both colliding particles move. According to the conservation of energy and momentum, we have 1 1 1 1 Ma (v a )2 + Mb (v b )2 = Ma (v ∗a )2 + Mb (v ∗b )2 2 2 2 2 Ma v a + Mb v b = Ma v ∗a + Mb v ∗b

(6.62)

with the masses Ma and Mb of the particles a and b, respectively; v a , v b are the velocity of both particles before the collision and v ∗a , v ∗b are the velocities after the collision. A possible change of the internal energy of particles is not considered here. The relative velocities are v r = v a − v b and v ∗r = v ∗a − v ∗b ; their absolute values are given by  |v r | = vr = (vax − vbx )2 + (vay − vby )2 + (vaz − vbz )2  ∗ − v ∗ )2 + (v ∗ − v ∗ )2 + (v ∗ − v ∗ )2 . |v ∗r | = vr∗ = (vax (6.63) ay az bx by bz The velocities before and after the collision can be expressed using the relative velocities and the velocity of the center of mass vc . By definition of the CMsystem, the equation Ma (r a − r c ) + Mb (r b − r c ) = 0

(6.64)

relates the radius vectors of both particles r a , r b to the vector r c = r a − r b . Taking the derivative of (6.64) yields the velocity of the center of mass v c = dr c /dt (6.65) Ma v a + Mb v b = (Ma + Mb )v c with respect to the laboratory system (compare with (6.11)). Since the total momentum is conserved, the velocity v c does not change. Performing several transformations leads to Mb v r = v s + ua Ma + Mb Ma vb = vs − v r = v s + ub Ma + Mb Mb v ∗a = v s + v ∗ = v s + u∗a Ma + Mb r Ma v ∗b = v s − v ∗ = v s + u∗b Ma + Mb r va = vs +

(6.66)

with v c = (Ma v a + Mb v b )/(Ma + Mb ) and the velocities in the CM-system ua , ub , u∗a , and u∗b . Using these relations in (6.62) shows that the absolute values of the relative velocities as well as the absolute values of the velocities in the CM-system remain unchanged, i.e., vr = vr∗ , ua = u∗a , and ub = u∗b . In the CM-system, the velocity vectors are turned only by an angle α (Fig. 6.5)

6.1 Binary Collisions ρp

63

α

ψ

Fig. 6.5. Collision of two moving particles in the CM-system

∞ 

α=π−2 rmin

r2

ρp dr 1−

ρ2p r2

−

2U (r) µvr2

,

(6.67)

which is determined by the interaction potential. In comparison to (6.30), the velocity vo is replaced here by the relative velocity vr . Again, the problem of two colliding particles is reduced to the motion of one particle with reduced mass µ in the central force field U (r). In the case of Coulomb interaction, we have the familiar solution of (6.67) tan

α 2

=

C Z1 Z2 e2 = 2ρp Er 4πo ρp µvr2

(6.68)

since Er = µvr2 /2 (see (6.31)). It is convenient to describe the rotation of the relative velocity vector in a coordinate system where the z-axis is directed along the relative velocity before the collision v r (Fig. 6.6). Then, we have for

Vr *

Vb Vb

* Vr

Va α

Va*

∆Vr

Vr

α − 2

Vr*

α ∆Vr = 2V r sin − 2

Fig. 6.6. Change of the relative velocity vector during collision. Note that 2vr sin(dα/2) = vr dα

64

6 Particle Coupling

the velocity change ∆vrx = 2vr sin(α/2) cos(α/2) cos ϕ = vr sin α cos ϕ ∆vry = 2vr sin(α/2) cos(α/2) sin ϕ = vr sin α sin ϕ ∆vrz = −2vr sin(α/2) sin(α/2) = −vr (1 − cos α)

(6.69)

and obtain, together with (6.66), the change of velocity in that coordinate system (note v c =const.) for the particle “a” ∆vax = [Mb /(Ma + Mb )] vr sin α cos ϕ ∆vay = [Mb /(Ma + Mb )] vr sin α sin ϕ ∆vaz = −[Mb /(Ma + Mb )] vr (1 − cos α)

(6.70)

and correspondingly for the particle “b” ∆vbx = −[Ma /(Ma + Mb )] vr sin α cos ϕ ∆vby = −[Ma /(Ma + Mb )] vr sin α sin ϕ ∆vbz = [Ma /(Ma + Mb )] vr (1 − cos α) .

(6.71)

The angle ϕ is the polar angle in the (x, y)-plane and varies between 0 and 2π. The velocities after the collision are given by Mb ∆v r Ma + Mb Ma v ∗b = v b − ∆v r . Ma + Mb v ∗a = v a +

(6.72)

In the global coordinate system, which is fixed for all collisions, the different components of ∆v r are expressed as follows: ∆vrx = (vrx /vr⊥ )vrz sin α cos ϕ − (vry /vr⊥ )vr sin α sin ϕ − vrx (1 − cos α) ∆vry = (vry /vr⊥ )vrz sin α cos ϕ + (vrx /vr⊥ )vr sin α sin ϕ − vry (1 − cos α) ∆vrz = −vr⊥ sin α cos ϕ − vrz (1 − cos α) (6.73)   2 + v 2 and v = 2 + v 2 + v 2 . In the case of v where vr⊥ = vrx vrx r r⊥ = 0, ry ry rz we obtain, instead of (6.73), again (compare to (6.69)) ∆vrx = vr sin α cos ϕ ∆vry = vr sin α sin ϕ ∆vrz = −vr (1 − cos α) .

(6.74)

The equations of (6.72) and (6.73) can be directly used in particle simulation codes, when the motion and collisions of each particle are followed. General energy and angular relations for two-particle collisions together with the evaluation of the corresponding cross-sections are given in [97–101].

6.2 Particle Transport in Matter

65

6.2 Particle Transport in Matter The penetration of a particle, ion, or atom through solid material is accompanied by a partial or full loss of its electrons. At high velocities, the particle is fully stripped of its electrons. An overlapping of electron shells occurs resulting in a quantum exclusion of possible states. Target electrons can be captured by the particle. The effective charge of the moving particle is a function of its velocity and the kind of material. Conversely, the moving particle affects the target by polarizing the electron gas around it. Energy is transferred from the particle to the atomic positive nuclei, a process called nuclear stopping or elastic energy losses (Sect. 6.2.2), and to the target electrons called electronic or inelastic stopping (Sect. 6.2.3). Bohr concluded in his first papers that the energy loss of ions in matter can be divided into these two components. He estimated by applying simple recoil kinematics that the inelastic loss channel is the more important one. His reviews [102,103] provide up to now a very profound and clear physical description of the main processes and parameters. By losing its energy, the particle also changes its effective charge and, consequently, the interaction intensity with the target medium is influenced as well. This mutual dependence makes the analysis of particle penetration through matter difficult. Ziegler formulated in his comprehensive review of experimental data and fitting functions the main questions [104]: 1. How does an energetic charged particle lose energy to the quantized electron plasma of a solid? 2. How do you incorporate into this interaction the simultaneous distortion of the electron plasma caused by the particle? 3. How do you estimate the degree of ionization of the moving atom and describe its electrons when it is both ionized and within an electron plasma? 4. How do you calculate the screened Coulomb scattering of the moving atom with each heavy target nucleus it passes? 5. How do you include relativistic corrections to all of the above? It was hoped that by introducing the so-called effective charge of the projectile that the stopping power theories developed for proton penetration could be used for partially-stripped heavy ions. In the model of Bohr, all electrons of the particle with velocities lower than the instantaneous particle velocity are assumed to be stripped off. Lamb suggested a similar effective charge approximation, but based on the energy rather than on the velocity of the ion’s electrons [105]. Brandt and Kitagawa revised the Bohr suggestion of the ionization degree of ions traveling within solids and considered the velocity of the particle’s electrons with respect to the Fermi velocity of the solid [106]. Electrons with velocities smaller than the relative velocity of the ions of the Fermi velocity are stripped off. Their new concept has proved to be quite accurate. The Fermi energy of the medium can be measured in experiments of electron transmission through thin foils.

66

6 Particle Coupling

The first unified approach to stopping and range theory was given by Lindhard, Scharff, and Schiott [107,108]. The range of ions could be predicted within a factor of 2 for all species and velocities up to the velocity of the stopping power maximum. The best agreement could be achieved for ions, which are neither fully stripped nor nearly neutral [104]. Further improvements of the theory are based on numerical methods, which allowed for the consideration of more realistic interaction potentials. Stopping powers can now be calculated with an average accuracy of several percents overall. Range distributions for amorphous elemental targets have about the same accuracy. The stopping power of ions in compounds has been analyzed in detail in [91]. The described treatment is based on the Bragg rule, i.e., a linear superposition of the inelastic stopping powers of the constituents weighted by their concentrations is applied. Due to the complex bond structure in materials such as organic compounds, the determination of the correct excitation and mean ionization potentials remains a difficult task. 6.2.1 Definitions and Main Parameters Starting with an energy Eo , the total range of particles penetrating a medium is given by −1 0
dE dE . (6.75) Ro = dx Eo

Although, rather than being a stopping force, the rate of energy loss dE/dx, defined as the energy loss per unit distance x, is commonly called stopping power for historical reasons. The stopping power depends on the particle’s velocity as described in Sects. 6.2.2 and 6.2.3. While slowing down, a particle may change its initial direction of motion by elastic encounters with target nuclei. Three types of possible trajectories can be distinguished (Fig. 6.7). The penetration of particles through matter can be described by several parameters. The path length is the total length of the trajectory, the depth is the distance from the surface to the position where the particle is stopped, and the spread is the distance from the point of impact to the surface point

(a)

z

(b)

z

(c)

z

Fig. 6.7. Trajectories of particles for (a) σ ∗
1, (b) σ ∗  1, (c) σ ∗  1

6.2 Particle Transport in Matter

67

determined by projecting the point of rest onto the surface. In addition, we have the radial range, which denotes the distance from the point of impact to the point of rest in the material (Fig. 6.8). Furthermore, transverse and longitudinal projected ranges can be defined (see Chap. 10 in [109]). Introducing the ratio of the total range to the transport length as a new parameter allows for the classification of the processes of stopping and scattering Ro = Ro no σtr (6.76) σ∗ = ltr where the transport length ltr = 1/(no σtr ) is determined by the atomic density no and the transport cross-section  π dσ σtr = 2π sin θ dθ . (6.77) (1 − cos θ) dΩ 0 In the case of σ ∗ 1 (Fig. 6.7a), the particle moves on a rather straight trajectory, since no deflection by an angle of near unity occurs. This situation changes if σ ∗  1 (Fig. 6.7b), then at least, one strong deflection disturbs the straight motion. For σ ∗  1 (Fig. 6.7c), a more or less chaotic motion can be observed, which is described in the terminology of diffusion (random-walk) processes. The scattering parameter σtr gives the number of strong deflections along the total path. For non-relativistic electrons, we have approximately [110]
 −1 5.6 σ = 1 − exp − Z2 + 1 ∗



(6.78)

whereas for light ions in heavy media the parameter is about [110] σ∗ =

1/2

6.33M1

2/3 3/2

Z1 εo

ln(1 + 0.7εo )

(6.79)

with the reduced energy εo (see (6.88)). The atomic number of the target material Z2 is the key factor and largely defines the scattering behavior, i.e., the higher the ratio Z2 /Z1 , the more pronounced the scattering. Distributions are characterized by their moments. By taking only the first two moments into account, a good agreement with experimental results can often be achieved. Thus, the Gaussian normal distribution   1 (Rd − Rd )2 exp − , (6.80) f (Rd ) =  2 2 2∆R d 2π∆Rd here written for the depth distribution function, can also be applied for the energy distribution as well as for the distribution of scattering angles.

68

6 Particle Coupling Incident Ion x

Path Length Full Range

y Depth

Surface

Point of Rest

z Projected Range

Transverse Projected Range

Fig. 6.8. Schematic drawing for the definitions of depth and of the different ranges as indicated

The depth distribution function (6.80) gives the fraction of particles that have reached a depth of Rd where Rd is the mean depth and ∆Rd2 = (R − Rd )2 is the depth variance—the straggling parameter (Fig. 6.8). More sophisticated analytical distributions and theories of particle transport can be found in [107, 108, 111–113]. Today’s theoretical investigations are usually assisted by computer simulation techniques (see Sect. 6.10). As an example, the ranges and standard deviations are given for certain ion–target combinations and energies in Table 6.1. 6.2.2 Elastic Energy Loss During penetration, a particle has a large number of interactions with the nuclei in the target. In each collision, a part of its kinetic energy E∆ is transferred. These losses are called elastic energy losses and are defined as the loss of energy per unit distance dE = no Sn = no − dx

max E∆

E∆

dσ dE∆ dE∆

(6.81)

min E∆

where no is the atom density in the target and max E∆ 4M1 M2 = γk = Eo (M1 + M2 )2

(6.82)

gives the maximum energy. The maximum energy can be transferred in a elastic collision, namely during a head-on collision with α = π according to

6.2 Particle Transport in Matter

69

Table 6.1. Mean depth and standard deviation for normal incidence [109, 114]



Ion D D D D D D D D 4 He 4 He 4 He 4 He 4 He 4 He

Target C C C C Si Si Si Si Si Si Si Ni Ni Ni

Energy (eV) 10 102 103 104 10 102 103 104 102 103 104 102 103 104

Rd (nm) 0.3 2 20 160 0.7 4 25 200 2 14 100 2 7 45

∆Rd2 (nm) 0.2 1.2 9 40 0.4 2 13 65 2.5 10 60 0.8 5 20

(6.8); Sn is the stopping cross-section and dE/dx the stopping power. For Coulomb interactions (U (r) = Z1 Z2 e2 /(4πo r) = C/r), (6.81) can be readily calculated. For that purpose, we give the Rutherford cross-section (6.43) as a function of the transferred energy E∆ instead of the scattering angle α (see (6.8) and (6.17)) E∆ = E2 = 4Eo

M1 M2 α cos2 θ2 = Eo γk sin2 (M1 + M2 )2 2

(6.83)

since cos(θ2 ) = cos(π/2 − α/2) = sin(α/2) and dE∆ = Eo γk sin

α α cos dα 2 2

(6.84)

with the kinematic factor γk given in (6.82), and Eo = M1 vo2 /2 is the kinetic energy of the incoming particle. Substituting now sin(α/2) in terms of E∆ yields the desired result dσ πC 2 M1 1 = 2 dE∆ Eo M 2 E∆

(6.85)

since Er = Eo M2 /(M1 + M2 ), sin α = 2 sin(α/2) cos(α/2), and dΩ = 2π sin α dα. Using (6.85) in (6.81) leads to dE = no Sn (Eo ) = no − dx

max E∆

E∆ min E∆

max E∆ dσ πC 2 no M1 dE∆ = ln . min dE∆ Eo M 2 E∆ (6.86)

70

6 Particle Coupling

min It is obvious from (6.86) that the minimum energy to be transferred E∆ cannot be set to zero. The problem to define the limits always occurs when integrations are performed based on the Rutherford cross-section. The cutoff min corresponds to a maximum impact parameter, which can be chosen to at E∆ min appears in the logarithm, this equal to the interatomic distance. Because E∆ parameter is of marginal importance despite the uncertainty in determining a suitable maximum impact parameter. In practice, (6.81) does not allow an analytical presentation if realistic interaction potentials (Sect. 6.1.4) are considered. Often approximations fitted to numerical calculations are applied. It is helpful to use the normalized stopping cross-section

sn (ε) =

(M1 + M2 )o Sn (E) M1 Z1 Z2 e2 aL

(6.87)

as a function of the reduced energy ε=

aL M2 4πo E =E 2 EL e Z1 Z2 (M1 + M2 )

(6.88)

where aL is the Lindhard screening length (6.53). Together with the normalized range rρ x 4πM1 M2 no a2L rρ = =x , (6.89) RL (M1 + M2 )2 we have the relations −

dE EL dε EL no aL Z1 Z2 e2 M1 dε = no Sn (E) = − = sn (ε) = − (6.90) dx RL drρ RL o (M1 + M2 ) drρ

between the introduced parameters where sn (ε) = dε/drρ . Frequently used approximations are 0.5 ln(1 + 1.2288ε) √ ε + 0.1728 ε + 0.008ε0.1504 0.5 ln(1 + 1.1383ε) √ . sU n = ε + 0.0132 ε0.21226 + 0.19593 ε

sKr−C = n

(6.91) (6.92)

6.2.3 Inelastic Energy Loss The energy losses due to collisions with the target electrons are called inelastic energy losses. Especially at high energies of the particles, these losses dominate, since the light electrons are able to pick up considerable amounts of energy from the incident particle. The collisions with the nuclei cause mainly scattering and the energy losses (Sect. 6.2.2) are smaller. In accordance with the kind of model applied, the inelastic energy losses are called local, if the interaction with the bounded electrons in the clouds around the nuclei is considered [115, 116]. In other (non-local) models, the electrons are considered as

6.2 Particle Transport in Matter

71

homogeneously distributed in the target by forming an electron gas of constant density as a good approximation to the conduction electrons in metals. The perturbation caused by the charged particles are assumed to be small. The main parameter in the analysis is the ratio of the particle velocity to the Bohr velocity vo /vB . The latter value is equal to the velocity of the electrons in the first orbit of Bohr’s atomic model vB =

¯h e2 = = 2.18 × 106 m/s 4πo ¯h ao me

(6.93)

˚. A with the Bohr radius ao = 4πo ¯h2 /(me e2 ) = 5.29 × 10−11 m = 0.529 A proton with that velocity has an energy of about 25 keV. Three velocity regions can be distinguished. At small velocities of the incident particle, vo vB , the energy losses are negligible. With increasing velocity, the energy losses increase 2/3 as well and have their maximum at roughly vo = 3vB Z1 where Z1 is the atomic number of the incident ion. At higher velocities, the inelastic energy losses decrease, since the interaction time during the collisions becomes shorter (Fig. 6.9).

1. vo
vB The positive charge of an ion slowly penetrating an electron gas causes a redistribution of the electrons within the Debye sphere (Sect. 5.2). Due to inertia, the center of this negatively-charged cloud falls a little bit behind the

A C

D B

Fig. 6.9. Schematic presentation of energy losses as a function of the particle energy. (A) inelastic energy loss (B) elastic energy loss (C) Lindhard–Scharff formula (6.95) (D) Bethe–Bloch formula (6.100))

72

6 Particle Coupling

moving ion. This charge separation generates an electric field, which tends to slow down the ion. The energy loss of the ion as a consequence of the polarization action was found to be directly proportional to the ion velocity [117]. This is an analogy to the friction force acting on a moving sphere with radius r in a viscous fluid dE ∝ 6πηv vo r (6.94) − dx according to the Stokes relation where ηv is the viscosity. The effects caused by a charged particle penetrating a quantized electron plasma have been analyzed in [118]. Lindhard provided a full treatment in the case of non-relativistic interaction and suggested in [92] together with Scharff the relation −

Z1 Z2 v dE 1/6 = no SLS = 2Z1 e2 ao no = k Eo L 2/3 2/3 dx o (Z + Z )3/2 vB

(6.95)



7/6 1 Z1 Z2 dE eV Eo [eV] − , = 1.21 no 3 2/3 2/3 3/2 dx ˚ M A ˚ 1 [amu] A (Z1 + Z2 )

(6.96)

1

2

or

which well describes the experimental data of inelastic energy losses in the 2/3 velocity region vo < vB Z1 .

2. vo  vB At high velocities of the incident ion, i.e., vo  Z2 vB , the binding energies of the electrons and their own velocities can be neglected, i.e., the electrons of the target are assumed to be at rest. The corresponding cross-section for energy loss in binary collisions has already been derived in Sect. 6.2.2 and is (see 6.85) dσ 2πZ12 e4 = (6.97) 2 dE∆ (4πo )2 me vo2 E∆ with Eo = M1 vo2 /2, M2 = me , Z2 = −1, and v∞ = vo1 − vo2 = vo − vo2  vo . According to (6.81), for the energy losses along the path dE = no Z2 − dx

max E∆

min E∆

dσ 2πZ12 e4 Z2 E∆ dE∆ = no ln dE∆ (4πo )2 me vo2




max E∆ min E∆

 (6.98)

where Z2 no gives the number of electrons per unit volume. The maximum energy to be transferred in ion–electron collisions is well-defined as max E∆ = 4Eo

M1 M2 me  = 2me vo2 . 2 (M1 + M2 ) M1

(6.99)

6.2 Particle Transport in Matter

73

min However, the right choice of E∆ is questionable. It has been proposed to take min the excitation energy for E∆ , but the resulting energy losses would be too small. One could consider only collisions with an impact parameter smaller than the average distance between the electrons [119]. Darwin restricted his analysis to collisions with deep electron penetration [120]. Two decades later, Bethe [121] and Bloch [122, 123] derived their fundamental relation based on quantum mechanics for the stopping power of fast particles under the assumption that the oscillator strength can be represented by the ionization energy Iion . In addition, the Born approximation has been applied, i.e., the amplitude of the wave scattered by the atomic electron field has to be small, when compared to the one of the incident (undisturbed) wave. The problem with the factor of 2 comparing the results of classical and quantum mechanics calculations was resolved by Bloch by a proper treatment of the impact parameter in both models. It is worth noting that the wavelength of the particle is rather small in comparison to the interatomic distance. Thus, the classical treatment might be justified for most of the velocities of interest. In the non-relativistic case, the Bethe–Bloch relations states
 2me vo2 4πZ12 Z2 e4 dE = no SBB = no ln − dx (4πo )2 me vo2 Iion
 2me vo2 Z12 Z2 e4 ln = no . (6.100) 4π2o me vo2 Iion

For the mean ionization energy, several approximations are proposed such that −2/3 ), thereby trying to include the Iion = 13Z2 or Iion = 10.3Z2 (1 − 0.793Z2 contribution of excitation as well. Inherent to this treatment is that energies smaller than the ionization energy cannot be transferred. The Bethe–Bloch formula remains the principal tool to evaluate the energy loss of particles with velocities of 1 MeV/amu to 2 GeV/amu, of course taking relativistic corrections at higher velocities into account [104]. Several other coefficients such as shell corrections have been added to improve the formula [124]. Excellent reviews of relativistic particle (> 10 MeV/amu) stopping powers are given in [125] and [126]. For velocities below 1 Mev/amu, this relation fails because the ion projectile may not be fully stripped of its electrons as assumed in the theory. 3. vo  vB At the maximum of energy losses, the particle velocity is of the same order as the velocities of the electrons in the target. Partially-stripped particles, overlapping electron clouds, and the necessity of dealing with binary collisions in their general form are some of the aspects to consider. Despite the theoretical effort [118, 127–131], no satisfactory solution could be found. Up to now, approximations of the type [132] S=

SLS SBB SLS + SBB

(6.101)

74

6 Particle Coupling

where SLS is the stopping power as given by the Lindhard–Scharff formula (6.95) at low velocities and SBB is the Bethe–Bloch stopping power (6.100) at high velocities, are used in the intermediate energy region [104].

6.3 Material Modification by Ion Beams Comprehensive reviews on material modifications due to ion beam impact are given in [133–138]. The main processes are defect production and agglomeration, compositional changes of alloys by preferential sputtering, Gibbsian segregation, displacement mixing, bombardment-induced decomposition, radiation-enhanced diffusion, and phase transformation. All these processes are governed by a number of parameters, such as ion type and energy, target temperature, structure and composition, and ion flux density. Thus, a large variety of structures and composition distributions can result from ion bombardment of solid compounds. Preferential sputtering is triggered by differences of mass and surface binding energy. Atoms of different elements are emitted at different rates. The resulting compositional changes in the top atom layer are accompanied by segregation. This effect, often called Gibbsian adsorption, tends to minimize the surface free energy by increasing the concentration of one alloy constituent in the outermost atom layer relative to that in the bulk. Usually, segregation requires elevated temperatures, where thermally activated diffusion can occur but can also be bombardment-induced. During irradiation, a number of trajectories are terminated near the target surface and are followed by one or more low-energy, chemically-guided steps. The spatial separation between defect production and annihilation establishes defect fluxes, usually directed toward the surface. Since the motion of defects requires the motion of atoms, the fluxes of alloying elements couple to the defect fluxes. Ion bombardment provides sufficient energy leading to recoil generation, cascade evolution, atom mixing, and redistribution of constituents. Amorphization as a process of lowtemperature mixing is often observed. Defects are produced during collisions of ions and displaced atoms with other atoms with the result that bulk atoms receive an energy of Ed ≈ 25 eV, which is necessary to make an irreversible displacement out of a crystal lattice. Note that the displacement energy is not a well-defined quantity and ranges from 10–40 eV. It depends on the direction of the pushed target atoms, and how large its energy should be to form a stable Frenkel pair, i.e., a vacancy and an interstitial displaced atom. These are point defects. Based on the model proposed in [139], the number of produced Frenkel pairs NF can be estimated as (1 − RE )Eo (6.102) NF  0.8 2Ed where RE is the energy reflection coefficient. This estimation can be improved by including the elastic energy loss in the cascade and the energy loss due

6.3 Material Modification by Ion Beams

75

to transmission and sputtering [109]. These additional losses obviously reduce the energy at disposal for damage production. As a result of secondary processes such as diffusion and joining of point defects into clusters, more complex defects are formed: dislocation loops, vacancy pores, or, in the presence of a gas, bubbles. By ion irradiation, the material properties can be modified: (1) the yield stress of metals and alloys increases considerably at temperatures below about 0.35 of the (absolute) melting temperature leading to hardening and embrittlement, (2) swelling caused by the nucleation and growth of cavities occurs, and (3) plastic deformation of materials caused by the simultaneous action of irradiation and mechanical stress (irradiation creep) occurs as well as high-temperature embrittlement due to radiation-induced impurity segregation to grain boundaries [134,138,140,141]. Helium-containing materials often fail by intergranular fracture. Since thermal diffusion in metals occurs predominantly by the position exchanges between atoms and vacancies, the presence of excess defects during ion bombardment at elevated temperatures results in radiation-enhanced diffusion. The analysis of temperature-controlled processes such as diffusion and segregation is complicated owing to the dependence of the corresponding coefficients of temperature, irradiation dose, ion energy, the structure of the material, as well as on the particle flux density in the case of radiationstimulated effects. The theoretical study of the behavior of gases in materials is based on a set of transport equations that contain a large number of uncertain parameters such as trap binding energies, mobility, diffusivity, solubility, recombination coefficients, defect microstructure [142], and can, in the best of cases, explain the observed behavior. Especially in the case of graphite with its complex C–H bonding structure, even computer simulations are hampered due to the lack of reliable basic data [143–146]. Many of the listed processes, for example displacement mixing, can be described by the diffusion formalism by using effective diffusion coefficients, which are expressed in terms of jump rates and jump lengths (see (6.109)). The diffusion equation for constant, but anisotropic diffusion is given by m ∂2n ∂n  = Di 2 ∂t ∂xi i=1

d

(6.103)

where n(r, t) is the particle density for time t at the position r = (x1 , ..., xdm ), and dm is the dimension. The change of the particle density in time is, therefore, defined by the difference of the incoming flux into a volume element and the outgoing particle flux. The convective part of the particle flux (Γ = n vconvection ) is neglected in (6.103). If all particles N started to diffuse at t = 0 from the same position r = ro , then the solution of (6.103) yields  ! d
d " m m  x2i 1 √ n(xi , t) = N exp − . (6.104) 4Di t 4πDi t i=1 i=1

76

6 Particle Coupling

Hence, the surfaces of same particle density are spheres in the case of isotropic three-dimensional diffusion and ellipsoids y2 z2 x2 y2 z2 x2 + + = + + = const. 2νx λ2x t 2νy λ2y t 2νz λ2z t 4Dx t 4Dy t 4Dz t

(6.105)

if the diffusion is different in different directions. The distance between the position reached after time t and the initial position ∆xi = |xi − xio | is determined by the diffusion coefficient Di (if xio = 0) 1 (∆xi )  = N

∞

2

n(xi , t) x2i dxi = 2Di t

(6.106)

−∞

where for one-dimensional diffusion D = (∆x)2 /(2t). In the case of isotropic diffusion, i.e., D = D1 = D2 = D3 = Ddm , we have |∆r|2 =

dm 

(∆xi )2

(6.107)

i=1

and (∆r)  = 2

dm 

(∆xi )2  = 2dm Dt

(6.108)

i=1

with D=

1 (∆r)2  1 = νλ2 . 2dm t 2dm

(6.109)

Using (6.109), the diffusion coefficient can be readily determined in particle simulation codes. For each particle, the distance from a starting point has to be obtained after a certain time t, which should be much larger than the mean collision time (t  τ ), and then their squares have to be averaged for all particles simulated. Another method to obtain the diffusion coefficient is via 1 (∆r)2  D= (6.110) lim 2dm t→∞ t by calculating the square of the distance between initial and current positions over the simulation time, and averaging this for all particles. This curve shows for larger time a linear asymptotic behavior; its slope is directly proportional to the diffusion coefficient according to (6.110). In fusion experiments, erosion and powerful destruction events as well as deposition of material dominate the change of surface composition and surface structure. Not to forget, the rather large roughness of the used plasmafacing materials makes the discussion in terms of atom layers somewhat academic.

6.4 Retention and Tritium Inventory Control

77

6.4 Retention and Tritium Inventory Control In principle, four mechanisms can result in tritium retention: (1) direct implantation of tritium ions, (2) diffusion into the bulk, (3) production of tritium by transmutation nuclear reactions, and (4) codeposition of tritium with eroded material (Fig. 6.10). The amount of implanted fuel atoms in ITER can be roughly estimated at MT = mT cT/C nC Swall dimpl = 2.3 g

(6.111)

where mT = 5 × 10−27 kg is the tritium atom mass, cT/C is the concentration of tritium in carbon (here set to unity), nC = 11.3 × 1028 atoms/m3 is the atomic density of graphite, Swall  800 m2 is the area of all wall components in ITER, and dimpl  5 × 10−9 m [114] is the average implantation depth according to an energy of about 100 eV of charge-exchange neutrals. With lower impact energies and grazing incidence, the implantation depth becomes much smaller and the implanted amount is far below the ITER limit of 350 g. Diffusion into the depth of the bulk material, e.g., along grain boundaries, could lead to a higher inventory, but the resulting amount of trapped tritium does not reach critical values [147, 148]. On the other hand, codeposited layers with a thickness of several hundred micrometers are often observed in fusion experiments using carbon-based wall components. The ratio of tritium to carbon in such deposited layers depends on the surface temperatures, but can reach values up to 0.4, and in so-called soft layers up to unity. At temperatures of 1000 degrees centigrade, this ratio goes down to 0.1. No saturation effect is observed, i.e., as long as new carbon is sticking to the surfaces tritium will be codeposited. This codeposition mechanism is clearly the dominant one with respect to tritium retention. Plasma Chemisorbed Tritium

C

T Codeposition of T with C

Implanted Tritium

Implanted Carbon

Surface Diffusion

Pores Transgranular Diffusion

Material

Grain Boundary

Fig. 6.10. Schematic of different retention and diffusion channels of tritium [146]

78

6 Particle Coupling

Unfortunately, not only carbon materials but also materials such as beryllium and tungsten show significant ability to be codeposited with fuel ions [149]. Prevention of film deposition or ensuring high re-erosion of deposited films is a necessity. Experience from plasma technology research, despite its opposite aim of achieving high deposition rates, could be useful. Recently, experiments in ASDEX-Upgrade have shown that injection of nitrogen into the divertor plasma led to a strong decrease in the amount of deposited material without deteriorating the performance of the main plasma [150]. Which mechanisms are involved is a question of ongoing research. Chemical sputtering, surface chemistry effects, and gas-phase reactions are considered. Nitrogen bondings with itself or with carbon atoms are highly stable and lead to the formation of volatile products such as N2 , CN, HCN, C2 N2 . This process of chemical erosion is effective only at higher energies (E > 20 eV). It has been found that the evolving surface state influences the deposition efficiency. The higher the nitrogen concentration at the surface, the lower the growth rate. In addition, the bond structure at the surface affects the sticking of impinging species. The radical molecule as well as the ion chemistry in the plasma is significantly modified under the presence of nitrogenated species [150]. All these effects could help to inhibit film formation and, therefore, have the potential to reduce the tritium inventory due to codeposition. While oxygen is not a favored gas as far as tokamak operation is concerned, it is the basis for most techniques suggested to remove codeposited films [151]. Based on laboratory experiments on the oxidation of carbon films, which show a significant reaction of oxygen with the layers in the range between 500– 750 K, ventilation of TEXTOR with oxygen has been investigated to remove redeposited carbon material and to release the incorporated hydrogen. A significant part of the injected oxygen adsorbs on the walls due to the formation of stable oxygen compounds. Partly, oxygen reacts with carbon to form CO and CO2 [152]. The effect of thermo-oxidative removal of hydrogen from codeposited layers is not well understood. It is suggested that metal contamination could act as a catalyst, thus explaining the high erosion rates (a few µm/h) observed for the films, as compared to the graphite substrate [151]. While oxygen shows the highest removal rate, increased wall temperatures (e.g., 600 K) offer the possibility to remove codeposited layers by thermallyinduced chemical erosion with atomic hydrogen in ECR (electron cyclotron resonance) discharges [153]. ECR cleaning experiments have been performed also in Alcator C-Mod. Only localized cleaning has been observed with a rather low effective sputtering yield [154].

6.5 Impurity Generation Large energy and particle fluxes are deposited onto wall elements in fusion experiments. Owing to this intense plasma–wall interaction, material is eroded

6.5 Impurity Generation

79

by several mechanisms. The emission of atoms consequently may lead to plasma contamination. The main erosion processes are physical and chemical sputtering, thermal and radiation-enhanced sublimation, erosion by arcs, and blistering. Furthermore, there are exotic erosion effects such as potential sputtering and Coulomb explosion events (Sect. 6.6). Almost all eroded particles leave the surface as neutrals in the ground state. 6.5.1 Physical Sputtering Physical sputtering is the emission of surface atoms due to the impact of energetic particles and is described by momentum transfer. Target atoms involved in the developing collision cascade leave the surface if their received energy exceeds the surface binding energy Es . The energy of the sputtered atoms is, therefore, reduced by Es before escaping. Usually, the heat of sublimation as measured for real surfaces is used for the surface binding energy. Since the energy distribution of the sputtered particles has a maximum, the assumption of a planar potential at the surface is justified. Physical sputtering may be influenced by the target temperature due to a modification of the heat of sublimation. The surface binding energy decreases with increasing target temperature [155]. The most advanced analytical description of physical sputtering described in terms of collision cascade formation is presented by Sigmund and co-workers [156–158]. A comprehensive review of physical sputtering with respect to both experimental and theoretical investigations is given in [159]. The crucial quantity is the sputtering yield Y =

number of emitted target atoms . number of incident particles

(6.112)

In contrast to the reflection coefficient for example, the sputtering yield cannot be seen as a probability since it can be larger than unity. The sputtering yield depends on the ion–target combination, the energy of the incident particle, and its incident angle. By using the differential sputtering yield dY 2 /dEdΩ, the angle and energy distribution of the sputtered atoms is characterized. Sputtering is a rather short event. Two depths can be distinguished. Most of the sputtered atoms come out of the first two atomic layers (≈ 5 ˚ A), whereas a layer with a thickness of 25–50% of the mean depth is involved in cascade formation and transfer of the energy from the projectile to the target atoms. Three regimes of sputtering can be separated: (1) the regime of few collisions, (2) the cascade regime, and (3) the thermal-spike regime. In the case of light ions and low energies, cascade formation does not occur, since only few target atoms are involved, and their trajectories reveal a strong anisotropy. In this case, the eight different mechanisms proposed by Behrisch [160] can be reduced to four main processes illustrated in Fig. 6.11 [109]. It is sufficient to consider only recoils of the first generation (primary knock-on atoms (PKA)) and of the second generation (secondary knock-on atoms (SKA)). These recoils are

80

6 Particle Coupling Backscattered Ion

Incident Ion

SKA

PKA

PKA

PKA

SKA

Fig. 6.11. Sputtering mechanisms for low-energy light ions. PKA: primary knock-on atom; SKA: secondary knock-on atom (see Fig. 12.1 of [109])

produced in the top atom layers by the ions crossing the surface during the impact and after being backscattered. For lower ion energies, the backscattering channel is more important, for higher energies most of the sputtered atoms are released during the impact of the ion. The higher the energy and the mass of the incident ions, the more atoms are affected and recoils of several generations occur. The developing cascade can reach the surface and target atoms are emitted. The thermal spikes are, in fact, high-density cascades originated by high-energy heavy ions leading to an energy deposition on a very short time scale in a rather small volume. High temperatures are the results of these spikes accompanied by thermal sublimation and shock waves. Crater formation at the surface has been observed in the experiments. As a rough estimation, the sputtering yield can be expressed as the ratio of energy Edep deposited in the outermost layer of thickness d to the surface binding energy Es Edep Y  (6.113) Es since the target atoms that have received the energy Edep have to overcome the surface potential. The energy is transferred in elastic collisions. According to (6.86) (6.114) Edep = no Sn (Eo ) d with the stopping cross-section Sn as a function of the ion energy Eo . Thus, the sputtering yield exhibits the same energy dependence as the stopping cross-section—low sputtering yield at low and high energies and a maximum A at reduced energies of ε = 0.1 − 1. Taking the values no Sn (Eo ) = 0.1 eV/˚ d = 5˚ A, and Es = 5 eV, we obtain a sputtering yield of Y = 0.1. According to [161–163], the yield of physical sputtering can be calculated for normal incidence by 
2/3 
2 Eth Eth o (6.115) 1− Y (E, α = 0 ) = Qy sn (ε) 1 − E E

6.5 Impurity Generation

81

where E is the particle energy, and sn (ε), the nuclear stopping cross-section, is to be taken as [164] sn (ε) =

0.5 ln(1 + 1.2288ε) √ ε + 0.1728 ε + 0.008ε0.1504

(6.116)

based on the Kr-C interaction potential with ε = EM2 aL 4πo /[Z1 Z2 e2 (M1 + 2/3 M2 )] = E/ETF and the Lindhard screening length aL = 0.04685/(Z1 + 2/3 1/2 Z2 ) nm. Z1 and Z2 are the nuclear charges, and M1 and M2 are the masses in atomic mass units of the incident particle and the target atom, respectively. There are some analytical expressions for the parameter Qy and the threshold energy Eth [165–167], but usually these values are obtained by fitting the relation (6.115) to experimental and/or simulation data (see Table 6.2). For energies below the threshold energy Eth , the sputtering yield is zero. This energy threshold naturally appears as a result of the surface binding energy. Eth should be larger than Es , at least. However, the energy has to be transferred from the incident ion to a target atom and the target atom itself should change its initial direction after collision and move to the surface. This requires at least one additional collision. Bohdansky suggested the following relations [161]  Es /(γk (1 − γk )) for M1 /M2 ≤ 0.2 (6.117) Eth = 2/5 for M1 /M2 > 0.2 8Es (M1 /M2 ) Table 6.2. Parameters needed to calculate the sputtering yield from (6.115) [168] Target – particle Be (Es = 3.38 eV) Eth (eV) ETF (eV) Qy C (Es = 7.42 eV) Eth (eV) ETF (eV) Qy Fe (Es =4.34 eV) Eth (eV) ETF (eV) Qy Mo (Es =6.83 eV) Eth (eV) ETF (eV) Qy W (Es =8.68 eV) Eth (eV) ETF (eV) Qy

H

D

T

He

Self-sputtering

13 256 0.07

13 282 0.11

15 308 0.14

16 720 0.28

24 2208 0.67

31 415 0.05

28 447 0.08

30 479 0.10

32 1087 0.2

53 5688 0.75

61 2544 0.07

32 2590 0.12

23 2635 0.16

20 5517 0.33

31 174122 10.44

172 4719 0.05

83 4768 0.09

56 4817 0.12

44 9945 0.24

49 533127 16.27

447 9871 0.04

209 9925 0.07

136 9978 0.1

102 20376 0.2

62 1998893 33.47

82

6 Particle Coupling

with the kinematic factor γk = 4M1 M2 /(M1 + M2 )2 (the power of (5/2) in the relation given in [161] is a misprint). Fitting experimental and calculated data for many ion–target combinations, the following approximations have been proposed [164] Eth =7 Es




M2 M1




−0.54 + 0.15

M2 M1

1.12 .

(6.118)

The dependence upon the angle of incidence α with respect to the surface normal can be calculated as the following [169]:


Y (E, α = 0) 1 exp fy 1 − Y (E, α) = (6.119) cos αmax . (cos α)fy cos α The values fy and αmax are often used also √ as fitting parameters or can be = Es (0.94 − 0.00133 M2 /M1 ) and estimated with the relations [168] f y  αmax = π/2 − aL n1/3 / 2ε Es /(γk E) where Es is the surface binding energy (heat of sublimation) in eV, n is the density of the target material in atoms/m3 , and γk = 4M1 M2 /(M1 + M2 )2 . The angle αmax corresponds to the maximum of the sputtering yield. The energy distribution of sputtered particles, which leave the surface predominantly as neutrals, is given by [157] E dY ∝ dE (E + Es )3−2mp

(6.120)

where E is the energy of the sputtered atoms. The maximum of the energy distribution is at Es /(2 − 2mp ). Often, the value mp = 0 for a hard sphere interaction potential, which is reasonable at low energies, is assumed [170]. However, better agreement with experimental and simulated data is found, if for the exponent mp of the power potential a value of 1/6 is taken [157, 170]. A cosine distribution for the angular distribution of sputtered atoms is the consequence of an isotropic flux of recoils in the solid. In order to calculate the sputtering yields due to the bombardment of the plasma ions onto the material surface, the energy and angular distribution of the impinging particles have to be considered. A twofold averaged sputtering yield, defined as the yield averaged over the energy and angle of the incident ions is obtained by double integration [171, 172]. In addition, the effect of sheath acceleration should be included, since the incoming ions, which are assumed to have a Maxwellian velocity distribution at the sheath entrance, change their directions and energies quite considerably when crossing the electric sheath above the surface. In addition to thermal energy flux, (∝ 2kB Ti ) the impurity ions of charge state q obtain an energy of about q|eφw |  q 3kB Te in the sheath. The energy of (singly-charged) ions is then roughly five times the electron temperature for Ti = Te (see Sect. 8). The resulting average yield is then a function of the plasma temperature and the charge state of the ions

6.5 Impurity Generation

83

Fig. 6.12. Sputtering yields as a function of the plasma temperature for different ion–target combinations as indicated in the figure [173]

(Fig. 6.12). In contrast to (6.115), the threshold character disappears, as shown in Fig. 6.12, since there are always particles in the tail of the Maxwellian distribution with energies exceeding the threshold energy for sputtering as small as the plasma temperature might be. Sputtering yields obtained with a Monte Carlo simulation program are given in Table 6.3. A useful approximation of sputtering yields for fusion application can be found in [174]. Of course, the transformation of the velocity distribution in

84

6 Particle Coupling

Table 6.3. Calculated sputtering yields for various ion–target combinations obtained with the simulation program TRVMC, a version of the TRIM code [172] where a shifted Maxwellian distribution was used as the velocity distribution of the incident ions Yield YD+ →C YD+ →Si YD+ →Mo YD+ →W YC3+ →C YC3+ →Si YC3+ →Mo YC3+ →W

Te = 5 eV 2.4 × 10−4 2.5 × 10−4 − − 6.1 × 10−3 4.8 × 10−2 1.2 × 10−2 2.45 × 10−4

Te = 10 eV 3.7 × 10−3 4.0 × 10−3 1.0 × 10−6 − 3.7 × 10−2 1.4 × 10−1 7.1 × 10−2 1.6 × 10−2

Te = 20 eV 1.3 × 10−2 1.5 × 10−2 1.9 × 10−4 7.0 × 10−8 9.9 × 10−2 2.7 × 10−1 1.8 × 10−1 7.2 × 10−2

Te = 40 eV 2.1 × 10−2 2.6 × 10−2 3.1 × 10−3 5.4 × 10−5 1.9 × 10−1 4.1 × 10−1 3.2 × 10−1 1.6 × 10−1

the sheath can only be tackled to its full extent within the scope of kinetic simulations. The erosion of material exposed to a plasma, containing impurities, such as carbon shows a complicated non-linear behavior. This was demonstrated using computer simulation for the ion bombardment of various target materials [175]. Transitions from an erosion phase to a deposition phase and also reversed situations have been observed [172, 176]. Even under steady state plasma conditions, the erosion of an exposed target changes during the bombardment due to the change in surface composition. In the case of thin layers, wall conditioning layers for example, the lifetime depends on the kind of substrate material [177]. Owing to the higher reflection probability of plasma ions for high-Z materials, substrates made of such materials show higher erosion of the top layers. The general complexity arising in multi-component systems will be the subject of further investigation by experiments and numerical simulations in the near future. While analytic approaches are able to show dependencies and main trends more clearly, computer simulations provide detailed information based only on a very limited set of assumptions regardless of complexity of the geometry, material structure, and composition (Sect. 6.10). With respect to ion–solid interaction, two methods using particles can be distinguished: (1) the binary collision approximation, and (2) the molecular dynamics method. Both techniques are discussed in [109] where also a large bibliography of simulation studies is given. The sputtered particles leave the surface predominantly as neutrals. The small fraction of ions emitted depends strongly, however, on the electronic surface conditions via the work function. Oxygen tends to increase the positive fraction, and alkaline metals the negative fraction. The positive fraction increases with increasing velocity of the sputtered atoms. A similar statement applies to the excitation state of emitted target atoms, since ionization can be regarded as the highest excitation state.

6.5 Impurity Generation

85

6.5.2 Chemical Erosion Chemical erosion is characterized by the formation of volatile molecules during chemical reactions of the incident particles with the target atoms. Chemical erosion (or sputtering) is important in case of carbon-based materials under the bombardment with hydrogen and oxygen leading to an emission of a wide spectrum of hydrocarbon molecules and carbon oxides [178–180]. Since oxygen atoms react chemically with carbon and form CO and CO2 at a total yield of almost unity, a reduction of the oxygen level in the plasma will thus decrease the carbon contamination. The complex chemistry of the C–H system has been partly resolved by K¨ uppers and co-workers [181–186] regarding the thermal chemical reactivity. During the bombardment with hydrogen isotopes, the carbon atoms in the implantation zone are hydrogenated and form a complex C–H bond structure there. With increasing surface temperature, radicals such as CH3 are released, while at temperatures above 600 K the recombination of hydrogen to H2 prevails leading to a reduction of the erosion yield. These temperaturecontrolled processes are supported by radiation damage owing to the energy transfer from the incident ions to the atoms in the lattice. Thus, open bonds for hydrogen attachment are provided, if the ion energy exceeds a certain threshold. At low surface temperature, thermal release of hydrocarbons is not possible. However, the binding energies of the radicals (≈ 1 eV) at the surface is much smaller than the sublimation energy of carbon (7.4 eV) and they can be released by a bond breaking mechanism induced by ion impact. This process is called ion-induced desorption of hydrocarbon radicals [187,188] and was responsible of a still significant erosion yield at room temperature and low-impact energies as observed in the experiments. By combining thermal and ion-induced effects as described above, a semiempirical relation has been deduced and adjusted to available experimental data ([189] and personal communication with J. Roth (2000)) Ychem (E) = (Y1 + Y2 + Y3 )/4 + (Y4 + Y5 )/8

(6.121)

Yi = Yisurf + Yitherm (1 + Cd · Y damage ) Y damage = Qy sn (ε)[1 − (Ethd /E)2/3 ](1 − Ethd /E)2

(6.122)

with (i=1–5)

si Qy sn (ε)[1 − (Eth /E)2/3 ](1 − Eth /E)2 1 + exp[(E − 65)/40] 0.0439 si exp(−ci /Ts ) = 2 · 10−32 Γ + exp(−ci /Ts )

Yisurf = Yitherm

(6.123)

with si =

1 1+3· exp(−1.4/Ts )

2 · 10−32 Γ + exp(−ci /Ts ) × 2 · 10−32 Γ + [1 + 2 · 1029 exp(−1.8/Ts )/Γ ] exp(−ci /Ts ) 107

(6.124)

86

6 Particle Coupling Table 6.4. Parameters for Ychem (E) Parameter – particle Qy Cd Eth (eV) Eths (eV) Ethd (eV)

H 0.035 250 31 2 15

D 0.1 125 27 1 15

T 0.12 83 29 1 15

Fig. 6.13. Yields of chemical sputtering of graphite as a function of the target temperature and energy according to (6.121) for an ion flux density of Γ = 1022 1/(m2 s)

and c1 = 1.865, c2 = 1.7, c3 = 1.535, c4 = 1.38, c5 = 1.26. The ion flux density Γ is given in ions/(m2 s); the energy of the incident ions E and the surface temperature Ts should be inserted in electron volts. If the energy E is smaller than the threshold energy Eths , then the contribution of the surface process is zero (Yisurf = 0). The yield Y damage should be omitted if E < Ethd . The parameters Cd , Qy , Eth , Eths , and Ethd are given in Table 6.4. The nuclear stopping cross-section sn (ε) is given by (6.116). Using (6.121), yields of chemical erosion by deuterium ions with different energies have been calculated (Fig. 6.13). To call chemical erosion of carbon-based materials a complex process is an understatement. It depends not only on the ion energy, surface temperature, and ion flux, but also on the structure and composition of the particular graphite material. Small additives (dopants) such as boron or metals (Si, Ti) change the erosion yield considerably [190–192] owing to their long-range effect on the electronic structure in the C–H network. The observed reduction of chemical erosion at higher flux densities [193– 197] is a favorable effect but still not completely understood. It has been

6.5 Impurity Generation

87

suggested that time-consuming rearrangements of chemical bonds in the hydrogenation process of carbon atoms to hydrocarbon radicals as the precursor for sputtering might be the reason for the flux dependence [187]. A dependence of Γ −1 , observed in some experiments, results in a constant amount of eroded material in spite of a growing flux of incident ions. Using Bayesian probability analysis, a number of experimental data provided by several fusion experiments has been fitted and the following dependence has been suggested [198] Ychem (E, Ts , Γ ) =

Ychem,low flux (E, Ts ) 1 + {Γ/(6 × 1021 [1/(m2 s)])}0.54

(6.125)

yielding a decrease in the erosion yield with Γ −0.54 at high ion fluxes. The erosion yield at 1024 1/(m2 s) is expected to be only Ychem ≤ 0.005 [199]. Synergistic effects naturally arising in a plasma that consists (besides hydrogen) of different ions such as carbon, oxygen, and noble gases (such as argon, neon, and helium) play a significant role [200, 201]. These heavier impurity ions, often injected for radiation enhancement (see Sect. 9.1.5), increase the chemical erosion of carbon by an order of magnitude owing to efficient C–H bond breaking in the material [202]. The released hydrocarbons cover a wide spectrum of different Cx Hy molecules, the composition of which depends sensitively on the ion energy and surface temperature. The type of released hydrocarbon is decisive for further transport and reactions in the plasma. Despite the progress achieved over the last years in describing basic processes of chemical erosion by means of molecular dynamics (MD) simulations [203–207], this method suffers from still insufficient computer capabilities that restrict the calculations to very short time scales, far too short for a realistic description. Because of the lack of knowledge, constant chemical erosion yield of about Ychem = 0.01 − 0.02 and a fixed spectrum of released hydrocarbons are usually assumed in impurity codes such as the WBC [208] or ERO code [209, 210] for prediction of erosion and redeposition in fusion experiments. Reviews on chemical erosion can be found in [4, 6]. 6.5.3 Radiation-Enhanced Sublimation Enhanced erosion yields have been observed for ion bombardment of carbonbased materials at target temperatures above 1200 K. This effect is called radiation-enhanced sublimation (RES) and is related to diffusion and sublimation of interstitials created by the ion impact. This occurs when the transferred energy exceeds a threshold (displacement energy) which is about 25–35 eV for carbon. The mobility of interstitials in graphite is higher (migration energy 0.3–0.8 eV) as the vacancy mobility increases (migration energy 3.5–4.5 eV). The migrating interstitials may recombine immediately at the end of the socalled collision chain, some of them migrate freely until they either recombine

88

6 Particle Coupling

with vacancies, dislocations, grain boundaries or they agglomerate into interstitial clusters. After arriving at the surface, the interstitials are assumed to be thermally desorbed due to the low binding energies [211]. The yield of RES is determined by the number of interstitials produced in the surface layer with the thickness of the average diffusion length of interstitials. Interstitials originated in deeper layers do not have a chance to reach the surface and are annihilated at internal defect structures. The RES is particularly high at low flux density [212, 213]. It could be shown that the energy dependence of physical sputtering according to (6.115) can also be applied for RES, using the same threshold energy but a modified value for Qy [6]
QRES = Qy + 54M11.18 exp[−ERES /kB Ts ]

Γ 1020

−0.1 (6.126)

with the surface temperature Ts , the ion flux density Γ in ions/(m2 s), and ERES =0.75–0.85 eV. The decrease of RES with increasing ion flux, i.e., increasing interstitial production, is predicted to go as Γ −0.25 , but in experiments a weaker dependence on Γ −0.1 has been found. The carbon atoms are emitted according to the Maxwellian energy distribution characterized by the surface temperature and a cosine angle distribution. The emission of clusters (C2 , C3 ) is significantly reduced compared to thermal sublimation. Below 600 K, the erosion is dominated by physical sputtering. In the case of hydrogenic impact, chemical hydrocarbon formation occurs between about 600 and 1200 K. RES dominates above 1200 K where the erosion yield increases monotonically until it exceeds at 2000 K the physical sputtering yield by more than a factor of 10 [211]. Above 2800 K, thermal sublimation sets in. Radiation-enhanced sublimation by neutron bombardment has been found to be one or two orders of magnitude larger than the physical sputtering yield of neutrons due to momentum transfer in collision cascades. The summarized erosion by neutrons is about three orders of magnitude smaller than initiated by ions and charge-exchange neutrals [214]. 6.5.4 Thermal Evaporation With rising surface temperatures, more and more atoms are able to leave the target. The resulting evaporation flux density is determined by the vapor pressure p as a function of surface temperature Ts Γsubl = s

sp p(Ts )[P a] atoms n¯ v =√ (6.127) = s 2.6 × 1024 4 2πM2 kB Ts M2 [amu]Ts [K] m2 s

using n = p/kB Ts and v¯ = 8kB Ts /(πM2 ). The sticking coefficients for metals are about s =0.6–0.9 for graphite, which emits C, C2 , and C3 particles. The coefficient s is in the range of 0.05 for a C3 cluster and 0.4 for C atoms [166].

6.5 Impurity Generation

89

The vapor pressure p(Ts ) can be described in terms of the heat of vaporization (or sublimation) Es
 Es p(Ts ) = po exp − (6.128) kB Ts where kB is Boltzmann’s constant. Under steady state conditions the vapor pressure is determined by the equilibrium between sublimation and recondensation at the surface. A comparison of different materials with respect to sublimation can be found in [215]. For carbon, the following relation is useful C = Csubl · (Ts )g exp(−Es /kB Ts ) Γsubl

(6.129)

with g = 3.25, Csubl = 2.5 × 1020 K−3.25 /s, and Es = 7.42 eV [216]. Thermal sublimation and melting are practically unavoidable in fusion experiments at so-called “leading” edges of divertor and limiter plates, places where the magnetic field lines strike normally on the surface. The geometrical effect of heat load reduction due to grazing incidence is therefore nullified. 6.5.5 Blistering Blistering is the appearance of bubbles on the surface as a result of the formation of poorly dissolved gases. During bombardment with He and hydrogen ions, the formation of blisters at the surface has been observed at high fluences. This temperature-controlled effect occurs, when the He or hydrogen concentration in the material reaches values at which accumulation occurs, e.g., at grain boundaries it is energetically favored over interstitial positioning. The accumulation causes stress in the thin surface layer and can lead finally to flaking. From experiments with 1–15 keV He it is seen that the depth of maximum helium concentration is usually about a factor of 3 smaller than the cap thickness. Hence, the assumption of high gas pressure, which finally overcomes the mechanical strength, has to be modified. The effect of exfoliation can be explained by the formation of stress between the implanted surface layers and in the bulk material due to swelling. The stress finally leads to ruptures near the interface of the implanted and unimplanted region [217]. The critical fluence at which blisters start to appear has been found experimentally to be in the range of 1021 to 1022 He/m2 . Thickness of blister skins and size of blisters increase with energy. At a He energy of 3 MeV, the critical fluence for tungsten is about 0.5 × 1022 He/m2 with an average blister diameter of about 130 µm [218]. For hydrogen and deuterium the fluences are an order of magnitude higher. When the irradiation dose is increased the shells of the bubbles break and new blisters can form on the exposed surface. For low surface temperatures (0.1–0.2 Tm ), where Tm is the melting temperature, blistering occurs. For T < 0.4–0.5 Tm flaking prevails, and for T < 0.6 Tm blisters are formed with

90

6 Particle Coupling

dimensions that decrease as T is raised. Finally, for T > 0.6 Tm a porous surface appears, which is subjected to neither blistering nor flaking. The erosion rate is the highest for flaking. Fortunately, surface roughness and broad energy and angular distributions of incident ions can mitigate or even suppress the blister formation. Recently, bubble formation has been observed on powder metallurgy tungsten irradiated by hydrogen [219], deuterium [220] and helium ions [221] at surprisingly low energy (< 100 eV) but with a high flux density (> 1022 1/(m2 s)). A threshold energy for bubble formation has been found to be about 15 eV for helium bombardment. Hydrogen blister formation has been detected only at surface temperatures below 950 K. The mechanism of bubble formation is still under discussion since simple models fail to reproduce the observations. The projected range of He ions, for example, is estimated to be less than 10 ˚ A. The He contained in the bubbles is suggested to be supplied from the surface by diffusion processes.

6.6 Charge Effects Already by approaching the surface, at a distance of about 0.5–1.5 nm, the incident ions extract electrons from the first atomic layers and recombine before penetrating into the material [222]. The reversed process, called contact ionization, occurs if the gas atoms have an ionization potential less than the work function of the wall material. Then the gas atoms will be ionized upon contact with the surface. Thus cesium vapor (ionization potential 3.9 eV) ionizes when getting in contact with a tungsten surface (work function 4.5 eV). If the metal surface is heated at the same time to thermionic emission temperature, electrons will be emitted and a plasma is formed with the positive ions. This technique, used in Q-machines, establishes a highly-ionized, low-energy plasma free of instabilities, i.e., a quiescent plasma. Multiply-charged ions carry a rather large amount of potential energy and relax within times of about 10 fs via dielectronic processes in a transient “hollow” atom [223]. The electronic potential energy is transferred rapidly onto very small surface areas and associated with intense electronic excitation and emission of electrons, photons, and bulk atoms. The conversion of electronic excitations into motion of atoms, a process still under discussion, could lead to significant erosion yield and surface modification especially in the case of insulator materials. In contrast to the kinetically-induced sputtering process, this phenomenon is called potential sputtering [224,225]. Although of less concern in fusion experiments, potential sputtering holds great promise as a tool for more gentle nanostructuring of surfaces, i.e., modification of the topmost layers. Applications range from information storage via material processing to biotechnology [226]. An atomic cluster at speeds in excess of the Bohr velocity loses its binding valence electrons after transversing a few atomic layers, while retaining its

6.7 Diffusion-Controlled Sputtering

91

initial geometry. In addition to the accompanied electronic excitation of the target, the strong Coulomb repulsion among the ions of the fast cluster can result in a destructive event: the Coulomb explosion. The associated massive energy deposition finds applications in solid state detectors [227], material modification [228], and inertial confinement fusion [229].

6.7 Diffusion-Controlled Sputtering When an alloy is sputtered, the surface composition changes as a result of preferential sputtering of one of the constituent elements. In addition, diffusion processes affect the composition profile, the altered layer extending several atomic layers below the surface [230, 231]. In the simplest model of a binary alloy the equations of one-dimensional diffusion in the solid interior normal to the surface have to be considered. Assuming constant diffusion coefficients one has ∂ 2 cA ∂cA = DA ∂t ∂x2

and

∂ 2 cB ∂cB = DB ∂t ∂x2

(6.130)

where cA and cB are the number fractions of the “A” atoms and “B” atoms, respectively. By definition we have cA + cB = 1. Therefore, both equations are only consistent with DA = DB = D. Processes such as segregation, radiationenhanced diffusion, and ion mixing can be roughly described by introducing an effective diffusion coefficient. Experimentally, distances are measured relative to the actual surface. Let z be the depth of a point below this surface, which initially for t = 0 is at x = 0 (see Fig. 6.14). In general, the recession or deposition velocity v s (t) will not be constant, and we have t z =x−

v s (t ) dt

(6.131)

0

Initial Surface at t=0

Sputtered Layer x Surface at t >0

Surface Moving with vs

z

Fig. 6.14. Definition of the coordinates x and z

92

6 Particle Coupling

where v s (t) > 0 if the surface is receding. The diffusion equation for component A takes on the form ∂ 2 cA ∂cA ∂cA =D (6.132) + v s (t) 2 ∂t ∂z ∂z assuming the atomic density of the alloy remaining constant. Suppose that effective volumes 1/noA and 1/noB can be assigned to single atoms of A and B, which remain essentially constant through a range of alloy compositions. Then noA and noB denote the densities in atoms per unit volume of pure material A and B, respectively. Then, for NA and NB atoms in a total volume V of alloy one has V = NA

1 1 + NB o . noA nB

(6.133)

If nA = cA n and nB = cB n are the local densities (in atoms per unit volume) of the alloy constituents A and B, one obtains 1 = nA

1 1 cA n (1 − cA )n + nB o = o + o nA nB nA noB

(6.134)

with n being the density of the alloy n=

cA noB

noA noB . + (1 − cA )noA

(6.135)

Balancing all fluxes caused by erosion, reflection, deposition, and redeposition, B the change of surface concentrations ∆nA s and ∆ns (note, ns are given in 2 particles/m ) during a time interval ∆t is given by ∆nA s = ΓAero − ΓAdep − ΓAredep + ΓAref l ∆t redep dep = ΓAero [1 − fAredep (1 − RA )] − ΓAdep (1 − RA )

(6.136)

and ∆nB redep dep s = ΓBero [1 − fBredep (1 − RB )] − ΓBdep (1 − RB ) ∆t

(6.137)

is the number of atoms per unit surface element, f redep is the where nA,B s redeposition probability, and Rdep and Rredep are the reflection coefficients for deposited and redeposited particles, respectively. Using (6.136) and (6.137), the recession or deposition velocity v s (t) can be derived v s (t) =

∆nA ∆nB s 1 s 1 s s + = vA (t) + vB (t) . o ∆t nA ∆t noB

(6.138)

It is worth noting that the sputtering coefficients depend, of course, on the and csurface . Firstly, the surface actual composition in the surface layer csurface A B

6.8 Backscattering

93

binding energy is changed with composition [159, 232, 233]. Furthermore, the sputtering yield depends on the composition, since the momentum transfer is changed. Particularly, very light elements show up with large escape depths, larger than the usually assumed two or three atomic layers. In a first approximation, the assumption of linear correlation between the partial sputtering partial c = csurface Yi→j . yield and surface concentration can be made, i.e., Yi→j j Since the sputtering and deposition processes are mainly localized in the first atomic layers of the alloy, they can be used to define the boundary condition for the diffusion equation, which describes the evolution of the composition of much deeper layers. At the surface, we have
  ∂cA  s o s + v (t)cA n  vA (t)nA = Dn  ∂z surface
  ∂c  B s + v s (t)cB n  vB (t)noB = Dn (6.139)  ∂z surface

The first term on the right side (if taken as negative) is the diffusion flux entering the thin surface layer. The second term is due to the receding/growing surface. Changes of the alloy density due to diffusion can be neglected. The relations (6.132), (6.138), and (6.139) together with the conservation relation cB = 1 − cA constitute a self-consistent set of highly non-linear equations [230, 231] with the initial condition of a given composition profile at t = 0. A coupling of this mixing model with a plasma simulation code is required to analyze consistently the non-linear behavior of the plasma–wall interaction. redep and the fluxes of ions to the surface can be obtained The coefficients fA,B using the plasma code and serve as input parameters in the mixing model. In turn, the impurity concentration in the plasma is mainly determined by the eroded amount, while the “effective” sputtering yields YA,B→A,B depend sensitively on the current surface composition. The relative importance of the main processes involved (erosion/deposition on one hand, diffusion on the other) can be estimated by means of a dimensionless number, the Peclet number used in fluid mechanics: Pe = v∆z/D. The term ∆z is some characteristic length, which is in our case the thickness of several atomic layers, since most of the sputtered atoms originate from a very thin region at the surface. The “convective” velocity v is here the recession/deposition speed v s .

6.8 Backscattering Some particles can eventually leave the surface after traveling in the material. They are backscattered in a certain range of energy and angle with a probability given by the backscattering (or reflection) coefficient. In the experiments,

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6 Particle Coupling

the reflection of impurities can often not be distinguished from sputtering of the same impurity at the surface. Almost all reflected particles are neutrals. Backscattering is characterized by a broad energy spectrum of the reflected particles—energies from nearly zero to values close to the initial energy are found. The probability that a particle impinging with energy Eo under a direction (θo , ϕo ) is reflected with an energy E (E ≤ Eo ) into a solid angle Ω = (θ, ϕ) is given by the differential reflection coefficient d2 R(Eo , θo , ϕo ; E, θ, ϕ)/dEdΩ. This coefficient gives the probability for a particle to make the transition from a state (Eo , Ωo ) before incidence to the state (E, Ω) after backscattering. The total reflection coefficient of particles is given by  Eo RN =

d2 R(Eo , θo ; E, θ, ϕ) dE dΩ . dEdΩ

(6.140)

Ω 0

The energy reflection coefficient gives the fraction of energy carried away by the reflected particles

RE =

1 Eo

 Eo E

d2 R(Eo , θo ; E, θ, ϕ) dE dΩ . dEdΩ

(6.141)

Ω 0

The theoretical description of backscattering started with models of one strong deflection, e.g., for electrons [234–236] and for ions [237, 238], which are applicable at higher incident energies (see Sect. 6.8.1). In fact, one of few quantitative surface analysis techniques, the Rutherford backscattering method (RBS), is based on this simple one-collision model. The two-collision models available [239–241] help to refine the RBS spectra analysis by reducing the background signal. For lower incident energies of the particles, the assumption of diffusive transport due to many collisions with small deflection can be used [242, 243] (see Sect. 6.8.2). As usual, the region of intermediate energies causes problems in the theoretical analysis. The most straightforward (and most intricate) way is the solution of Boltzmann’s transport equation by selecting appropriate collision cross-sections for the energy region of interest [112,244–256]. Often, the particle backscattering coefficient RN is obtained based on the calculated implantation distribution R(z) in the material [257, 258] 0 RN =

R(z) dz .

(6.142)

−∞

Cutting simply the depth profile calculated for an infinite material is of course only a very crude approximation of the boundary condition.

6.8 Backscattering

95

Fig. 6.15. Additional trajectories that appear after a thin layer is added on top of a half-infinite material. Elastic collisions are indicated with small circles; energy loss due to collisions with target electrons is represented by springs [260]

The correct boundary condition with respect to backscattering can be considered using the so-called imbedding method [259]. The trick here is to add (in mind) a thin layer on top of a half-infinite material of the same kind (Fig. 6.15). Since the reflection probability for such a system should be the same before and after this operation, an equation can be deduced by notifying that the sum of all additional processes in this thin layer is zero. The result is an equation for the differential reflection coefficient R(E, Ω) that can be solved analytically in certain cases (for electrons [261], for ions [260]). Chandrasekhar invented this method by studying the transmission and reflection of light in the earth atmosphere. 6.8.1 One-Collision Model Fast particles (σ ∗ 1) can be assumed to be backscattered in one strong deflection with a target atom. Along the way down to the position of collision, sometimes deep in the material, as well as on the path back to the surface (after the collision), the particle moves on a rather straight trajectory. Along this path of length l it loses its energy in collisions with electrons. The probability that a particle scatters in a certain depth into a solid angle interval dΩ is then given by d2 R(Eo , Ωo ; E, Ω) =

dσel (Et , α) no dzt dΩ dΩ

(6.143)

where α is the scattering angle, and no dz is the number of target atoms per unit square. The initial energy of the particle Eo is simply related to the energy of the particle E after leaving the surface by  zt   dE    E t = Eo −  dl  0

zt / cos θ

dl , in

  dE     dl 

E = Et − ∆Eel − 0

dl

(6.144)

in

where Et is the particle energy in depth zt just before collision and θ is the reflection angle with respect to the surface normal (Fig. 6.16). According to

96

6 Particle Coupling Incident Ion Eo Reflected Ion E

Inelastic Energy Losses zt θ

Elastic Collision

Et α

Fig. 6.16. Graphical depiction of the one collision model for normal incidence

(6.83), the elastic energy loss is ∆Eel = Et γk sin2 (α/2). With (6.144), the energy E is definitely linked to a certain depth zt . Specifying the law of inelastic energy loss, (dE/dl)in , zt and dzt can be expressed in terms of E and dE, respectively. Thus substituting Et , zt , and dzt in (6.143) yields the desired relation for the reflection coefficient d2 R(Eo , θo ; E, θ, ϕ)/(dEdΩ). Analytically, this operation can be performed only for a limited class of presentations of (dE/dl)in (Sect. 6.2). For simplification, equal scattering angles in the laboratory system and in the center-of-mass system have been assumed, valid for M1 /M2 1. 6.8.2 The Diffusion Model In the case of σ ∗  1, the motion of the particle in the solid becomes fully isotropic after traveling the distance of the transport length ltr . Reaching the depth zt = ltr cos θo (θo is the incident angle), the particles start to move in all directions until they are stopped in the material or have left the surface. The affected volume has the shape of a sphere and is defined by the total range Ro minus the transport length as its radius (Fig. 6.17). In this model [242], the particle reflection coefficient RN is simply given by the ratio of the spherical segment (6.145) VR = 2π(Ro − zt )2 (Ro − 2zt )/3 to that of the total volume of the sphere V = 4π(Ro − zt )3 /3, i.e., 1 Ro − 2zt VR 1 1 − 2ltr cos θo /Ro = = V 2 R o − zt 2 1 − ltr cos θo /Ro 
1 1 − 2 cos θo /σ ∗ = 2 1 − cos θo /σ ∗

RN =

(6.146)

6.8 Backscattering

97

Incident Ion

Reflected Ion Backscattered Cone

z t= l tr R o- l tr

α

Fig. 6.17. Geometry of the diffusion model for the case θo = 0

under the condition that Ro > ltr (1 + cos θo ) .

(6.147)

The particles should have enough energy to leave the surface. Interestingly, the ratio of the surface area section 2π(Ro − 2zt ) to the total surface area of the sphere 4π(Ro − zt )2 as well as the ratio of the characteristic lengths (Ro − 2zt )/(Ro − zt ) give the same result (6.146). The prefactor 1/2 has to be considered in the latter case, since half of the particles are moving in directions away from the surface. This rather coarse approximation works surprisingly well for keV electrons [242,243]. However, its applicability is linked to condition (6.147); the model is only valid for σ ∗ > 2 cos θo . For large values of σ ∗ , (6.146) predicts a maximum reflection probability of 1/2.

6.8.3 Approximations The main parameter determining backscattering is the dimensionless quantity [260, 261] σ ∗ defined in (6.76). For half-infinite materials a somewhat better adjusted parameter ∗ σR =

Eo /|dE/dl| Ro  ltr [1 − cos(π/2 − θo )] ltr [1 − cos(π/2 − θo )]

(6.148)

can be used where θo is the angle of incidence with respect to the surface ∗ ∗ = σ ∗ . The parameter σR normal. For normal incidence, i.e., θo = 0, then σR gives, on average, the number of strong deflections required to leave the target. ∗ , the reflection probability can then be estimated. It increases with Knowing σR ∗ . increasing σR

98

6 Particle Coupling

Simple approximations of the total reflection coefficient can be constructed ∗ ∗ in the form of RN,E = a [1−exp(−b σR )] using the backscattering parameter σR ∗ or RN,E = c exp(−d/σR ) where a, b, c, and d are fitting parameters. Analytical relations for RN and RE can be found in [256, 262, 263]. For light ions such as H+ , D+ , and He+ in the energy range from 0.01 to 100 keV, the approximation [256] ∗ ) (6.149) RN = 1 − exp(−0.0513 σR can be used. Other fitting relations are [264] RN = [(1 + 3.2ε0.34 )1.5 + (1.33ε1.5 )1.5 ]−0.67

(6.150)

RE = [(1 + 7.1ε0.35 )1.5 + (5.3ε1.5 )1.5 ]−0.67

(6.151)

with the reduced energy ε according to (6.88). Further relations can be found in the review on ion reflection by Eckstein [6]. In the energy range below 10 eV, any chemical affinity of the incident particle, for example hydrogen, to the target leads to a decrease of the reflection probability [265], since the energy of incidence becomes comparable to the surface binding energy which is in the eV range. This effect is usually not included in the approximations. Despite the significant progress in the quantitative description of backscattering over the years, the powerful methods of computer simulation on fast computers available today have in fact terminated further theoretical development. Advanced programs such as TRIM [266–269] and MARLOWE [270,271] provide comprehensive information with a high degree of accuracy in all details. The quality of the results depends almost solely on the appropriate choice of the interaction potential. In the case of complex structures, geometries, and material mixture, computer simulations reveal their overwhelming power.

6.9 Electron Emission Due to the interaction of the ionic crystal lattices with the free moving conduction electrons, an electric field arises near the surface confining the electrons inside the material. This field has to be overcome by those electrons leaving the surface. At room temperature, the electrons are captured in the material. Electrons can be emitted by providing them with sufficient energy to overcome the surface potential barrier. Alternatively, the barrier may be modified, thereby increasing the probability of escape of high-energy electrons. Electrons may be emitted on account of electron collisions (secondary electron emission (SEE)) (Sect. 6.9.1), or collisions with positive ions, neutrals, or metastables. They can also be emitted due to photon absorption, surface heating (Sect. 6.9.2), or the application of an electric field (field emission) (Sect. 6.9.3). The photoelectric ejection of electrons, where the energy of the

6.9 Electron Emission

99

incident photons must be larger than the surface work function, is a minor effect in fusion experiments. Electron-induced secondary electrons make up the major part of electron emission. Fast positive ions induce electron emission from surfaces by intense local heating. Low-energy ions cause electron emission in a two-step process. First, they capture an electron from the surface thereby becoming neutralized in an excited state. Then, the excitation energy is transferred to a second surface electron, which can escape if its energy is larger than the surface work function. The electron emission coefficient for slow ions is considerably smaller than unity. No electron emission is expected due to neutral particles with thermal energy. However, fast neutrals can initiate electron ejection. This effect is used to detect neutral particles. Excited metastable neutrals produce secondary electrons due to the same mechanisms as explained for the ions. The helium metastable level, for example, is at 19.7 eV, while the work function of magnesium is 3.01 eV. Thus, secondary electrons are emitted with a maximum energy of 16.7 eV. Note that the total yield of electron emission might include a fraction of backscattered electrons. 6.9.1 Secondary Electron Emission (SEE) True secondary electrons have energies in the eV range. Traditionally, emitted electrons with energy lower than 50 eV are considered to be secondary, while electrons with higher energies are attributed to reflected electrons. The coefficient δSEE can be expressed in the following form [272] " !  E 2 E (6.152) exp −2 δSEE = δmax (2.72) Emax Emax as a function of the electron energy E. Values of δmax and Emax for materials of interest in fusion devices are listed in Table 6.5. The yield of secondary electron emission increases with the angle of incidence θ (taken with respect to the surface normal) according to [274] δ(θ) = δ(0)/ cosγSEE θ

(6.153)

Table 6.5. Parameters for δSEE from [273] Target Be C Fe Mo W

Z 4 6 26 42 74

δmax 0.5 1.0 1.3 1.25 1.4

Emax (eV) 200 300 400 375 650

100

6 Particle Coupling

with γSEE = 1 for all materials with nuclear charge Z > 10. For light elements such as Be, γSEE is close to 1.3. 6.9.2 Thermionic Electron Emission The amount of electron emission in the case of heated surfaces depends on the number of electrons in the metal with thermal energies greater than the thermionic work function WT and results in a flux of
 2 4πme ekB WT 2 js = T exp − (6.154) s h3 kB Ts according to the Richardson–Dushman law with the constant 2 /h3 = 1.2 × 106 A/(m2 K2 ) . ARD = 4πme ekB

(6.155)

Often, the constant A is corrected by fitting (6.154) to experimental data. For −5 carbon, AC A/(m2 K2 ) and WTC = 4.5 eV [275], for tungsten the RD = 3 × 10 −6 following values can be taken AW A/(m2 K2 ) and WTW = RD  ARD = 1.2×10 4.5 eV. Small additives, for example a few percents of thorium in tungsten, can change the surface work function of certain materials significantly. The thermal (or thermionic) emission of electrons can be considered as an analogy to the vaporization of a fluid. Let us calculate the flux of electrons out of the electron gas above the metal surface toward the surface. In thermal equilibrium, this rate of condensation is equal to the rate of vaporization, i.e., emission of electrons. The number of possible states of electrons in the electron gas (in the solid as well as above the surface) is dNZ = 2

dpx dpy dpz h3

(6.156)

where the factor of 2 owes to the fact that for each triad of momentum (px , py , pz ) two different directions of the electron spin are allowed. The probability fP (E, Ts ) to be in a quantum state of energy E is defined by the Fermi–Dirac distribution fP (E, Ts ) =

1 1 + exp[(E − EF )/kB Ts ]

(6.157)

with the Fermi energy EF . The electron density is determined by setting dn = fP dNZ =

dpx dpy dpz 2 . h3 1 + exp[(E − EF )/kB Ts ]

(6.158)

Since the energy of a motionless electron in a vacuum Uo is by several kB To larger than the Fermi energy, the unity in the denominator can be neglected (Fig. 6.18). The concentration of electrons with momentum in the interval (dpx , dpy , dpz ) is then
 2 E − EF dn = 3 exp − (6.159) dpx dpy dpz . h kB Ts

6.9 Electron Emission

101

U

Vacuum Uo

WT EF Metal E cond

Fig. 6.18. Potential energy of an electron in a metal (WT : thermionic work function, EF : Fermi energy, Econd : minimum energy of the conduction band, Uo : energy of a motionless electron in a vacuum)

Having the z-axis directed into the metal, the current density of electrons to the surface with momentum in (dpx , dpy , dpz ) is given by djs = e

pz dn . me

(6.160)

To obtain the total current density, (6.160) has to be integrated over positive pz -values and over all px , py -values  
 ∞ ∞ ∞ p2x + p2y + p2z 2e Uo − EF js = 3 exp − exp − pz dpz dpx dpy . h me kB Ts 2me kB Ts 0 −∞ −∞

(6.161) Here, the energy as the sum of the potential and kinetic energy E = Uo +

1 (p2 + p2y + p2z ) 2me x

(6.162)

√ in the vacuum is used. The integration over px and py yields 2πme kB Ts since ∞ exp(ax2 )dx = π/a, while the integration over pz gives me kB Ts . Thus, −∞
 2 4πme ekB Uo − E F 2 js = T exp − (6.163) s h3 kB Ts as already given above (see (6.154)). The thermionic work function WT = Uo − EF (Fig. 6.18) is equal to the work required to transfer an electron (with the largest kinetic energy in the metal) out of the metal into a vacuum state with no kinetic energy. 6.9.3 Electron Emission by the Application of an Electric Field Schottky showed that a reduction of the effective work function by an applied electric field enhances the electron emission. This is a quantum tunneling

102

6 Particle Coupling

effect and requires fields on the order of 107 –108 V/cm. However, significant emission enhancement is found already at a field of about 106 V/cm. This is ascribed to surface roughness resulting in higher electric fields at the tips of small surface protrusions. Another explanation is the lowering of the effective work function due to surface contaminations.

6.10 Modeling of Particle–Solid Interaction Starting in the 1950s and continuing today, computer simulation has been an indispensable tool in the analysis of particle–solid interaction processes [109]. The following sections are dedicated to the two main classes of simulation techniques in this field—the so-called molecular (or classical) dynamics model (MD) (Sect. 6.10.1) and the Monte Carlo methods (MC) (Sect. 6.10.2). Short descriptions of the basic ideas are given together with a list of steps to be executed in a computer program. 6.10.1 Molecular Dynamics Wherever the assumption of binary collisions fails, the many-particle interaction should be adequately described. The motion of each particle of a system consisting of N particles is defined by the force Fi exerted on this particle by the rest of them, i.e., N − 1. Self-forces are not considered, i.e., a particle does not act on itself. Whether the interatomic forces are calculated by an empirical potential or by ab initio techniques, the method for moving a particle in the MD simulation is the same, just integrating the Newtonian equations of motion N  d2 r i (t) dv i (t) = F = M = F ij (6.164) Mi i i dt2 dt j=1,i=j

in time for each particle i consecutively. The forces are usually derived  from a potential energy U (r), i.e., F ij = −∇U (rij ), with the distance rij = r 2i − r 2j between the particles i and j. The advantage of tight-binding molecular dynamics over classical potential simulations is the explicit incorporation of the real electronic structure and bonding of the material. These data are evaluated by codes based on first principles. MD simulations are very expensive. Especially, the calculation of the interatomic forces is much more time-consuming than the integration of (6.164). The analyzed systems consist, therefore, of no more than a hundred to a few thousand particles. Thus, the system size, for example, a crystal lattice, is rather small in real dimensions. Usually, periodic boundary conditions are applied or the atoms at the border are simply fixed. The simulation starts with some initial configuration, which is then relaxed to find the minimum energy structure. To introduce temperature into the system, the velocity of each particle is scaled at every time step in the way that the total kinetic energy of

6.10 Modeling of Particle–Solid Interaction

103

the system is given by kB Ts /2 per each degree of freedom. The temperature of the system can also be coupled to an external heat reservoir allowing the temperature to become a dynamical variable. The integration of (6.164) over small time steps can be performed using various numerical methods, for example, by the Runge–Kutta method of different orders. However, low-order algorithms have some advantages: no storage of high derivatives of positions, velocities, etc., is necessary, and larger time steps can be used without risking the violation of energy conservation. In addition, the number of force calculations in each time step should be reduced as much as possible, in the ideal case of one force calculation per time step. There is a trend to favor the family of Verlet algorithms [276], and, in particular, the velocity Verlet algorithm [277] ∆t F i (t) 2Mi r i (t + ∆t) = r i (t) + ∆t v i (t + ∆t/2) ∆t v i (t + ∆t) = v i (t + ∆t/2) + F i (t + ∆t) . 2Mi

v i (t + ∆t/2) = v i (t) +

(6.165) (6.166) (6.167)

This scheme advances the velocities v and coordinates r over a time step ∆t, which is on the order of femtoseconds. The time step should be an order of magnitude less than the period of the fastest oscillation, for example, bond stretching takes about 10 fs. After step (6.166), the force F i (t+∆t) is calculated, knowing the new positions, for step (6.167). The Verlet scheme has the advantage of high precision, while exhibiting low drift, i.e., the total energy fluctuates about some constant value. Furthermore, the Verlet method is symplectic, which means that the system can be traced back by reversing the momenta of all particles. Nonsymplectic methods, such as the predictor–corrector schemes, usually have problems with energy conservation for longer simulation times. In general, numerical methods cannot accurately follow the true trajectories for very long times. The ergodic behavior of classical trajectories, i.e., the fact that nearby trajectories diverge from each other exponentially fast due to the Lyapunov instability, sets a limit. However, the averaged values remain unaffected. Simulations run typically 103 –106 ∆t steps, corresponding to a few nanoseconds of real time, only in special cases extending to microseconds. It is thus important to check, whether an equilibrium is reached in the simulation during these rather short times. Unfortunately, many physical processes of interest would require an analysis on much larger time and spatial scales. 6.10.2 Monte Carlo Methods This class of methods got its name simply because of the use of random numbers. In particle simulation, the trajectory of each single particle is followed in time. At each time step decisions have to be made, for example, whether

104

6 Particle Coupling

a collision occurs or not. When applying random numbers (generated by a pseudo-random generator), the code becomes surprisingly short and efficient. Of course, one could generate a whole crystal structure and keep track of the positions of all target atoms and of the projectile to decide which certain target atom will be the next collision partner. The effort is significantly reduced, if a fixed mean free path is assumed and the impact parameter for the next collision is diced out. Since most real materials are amorphous or polycrystals, this approach gives, on average, a quite realistic description. The final results are obtained by a statistical analysis of all trajectories. To calculate total quantities such as the particle and energy reflection coefficient, several thousand trajectories are sufficient. However, millions of trajectories have to be calculated, if one wants to know, for example, the angular and energy distribution of reflection d2 R(Eo , θo , ϕo ; E, θ, ϕ)/dEdΩ. Considering processes such as implantation, damage production, reflection, sputtering, and transmission requires a detailed knowledge of the penetration of charged particles through material. In contrast to analytical solutions of the Boltzmann transport equation, the MC methods have significant advantages: • Realistic interaction potentials can be used instead of inaccurate approximations. • Real surface geometries, inhomogeneities in structure such a cracks, pores, and fibers can be taken into account. • Arbitrary ion–material combinations are possible to consider multi-element compounds and alloys of any desired composition. • Information may be obtained, which is not accessible either to experimental investigations or to analytical studies. Regardless of the many options and variations, let us discuss here the main steps in performing a MC particle simulation. Every particle starts from a defined position, usually at the surface, with a certain initial energy Eo into a certain initial direction given by (θo , ϕo ). It changes its direction due to binary collisions with target atoms and loses its energy predominantly in collisions with electrons. The position of the next collision partner is determined either by saving all atomic positions of the crystal matrix (as done in the MARLOW code [270, 271]) or by evaluating the impact parameter using a random number (as done in programs such as TRIM [266–269] or ACAT [278]). The latter approach is justified, when the surface affected by the particle beam is much larger than the grain size of the material. The following steps should be programmed: 1. Definition of the ion–material system (Z1 , Z2 , M1 , M2 , no ). 2. Definition of the initial energy Eo , direction Ωo = (θo , ϕo ), and position. In the case of crystal materials, the point of incidence at the surface can be varied by using two random numbers. −1/3 3. Each particle moves in steps of the mean free path lp = no . Thus, the distance between two collisions is just equal to the interatomic

6.10 Modeling of Particle–Solid Interaction

distance. Moving along this path, the particle loses energy in collisions with electrons L ∆Ein =

dE dE dE dl  lp = n−1/3 dl in dl in dl in o

(6.168)

0

where (dE/dl)in is the inelastic energy loss per unit length (see Sect. 6.2.3). The simple model of constant mean free path turns out to be very effective, and accurate. In other models the distance to the next collision is evaluated at each step. For this, one has to define the minimum deflection angle and to consider in some way the energy dependence of scattering and energy loss. This can become a problem in the case of a very long mean free path. 4. At the end of lp , a collision occurs. The impact parameter ρp is determined with a random number RND, which is distributed uniformly in the unit interval [0, 1]  RND . (6.169) ρp = 2/3 πno 5. The scattering angle α is calculated in the center-of-mass system (Sect. 6.1.1) and then transferred into the laboratory system tan θ1 =

sin α cos α + M1 /M2

or (since sin2 (α/2) = (1 − cos α)/2) 2 C(1 − C) tan θ1 = 1 − 2C + M1 /M2

(6.170)

(6.171)

with C = sin2 (α/2). The new direction vector is given by the cosine law of a spherical triangle cos θi = cos θi−1 cos θ1 + sin θi−1 sin θ1 cos(ψ)

(6.172)

where cos θi−1 is the angle with respect to the surface normal before and cos θi the angle after the ith collision (Fig. 6.19). The azimuthal angle ψ = 2πRND is calculated using another random number. 6. The energy loss in collision with the target atom is calculated by (6.83) ∆Eel = 4E

α M1 M2 sin2 (M1 + M2 )2 2

(6.173)

and subtracted from the current particle energy, i.e., E = E − ∆Eel .

105

106

6 Particle Coupling z

θi - 1 θi

γ

ψ α

vi - 1 vi

y

ϕi - 1 ϕi γ x

Fig. 6.19. Change of the velocity vector in the ith collision

7. A control of energy and position is performed. If the energy E becomes smaller than a so-called cutoff energy (usually on the order of the binding energy of several eVs), the particle is stopped. While staying in the region of modeling, i.e., as long as the particle has not left the surface, the motion of the particle is followed further by repeating steps 3 through 7. The listed steps are repeated many times, for many particles. This is followed by a statistical analysis. The implantation profile, for example, is obtained simply by saving the end position of all particles. In analyzing sputtering phenomena, the knock-on target atoms, which obtained from collisions sufficient energy to leave their position in the lattice, have to be followed in the same way as done for the projectile. The simulation of cascade formation and development demands only a slight increase of the coding effort. At the surface, the target atom has to overcome the surface potential. This results in a change of its emission direction due to refraction in this planar potential (Sect. 6.2.4 in [109]). In (6.169), a random number RND is used to model the distribution of all possible impact parameters f (ρp ). We have ρp RND =

f (ρp ) dρp ,

√ ρp = ρmax RND , p

(6.174)

0

since the differential cross-section is dσ = 2πρp dρp . We obtain f (ρp ) = (dσ/dρp )/σtotal = 2πρp /(πρ2p ) .

(6.175)

For a cylindrical volume of length lp , the maximum impact parameter which corresponds to the smallest scattering angle is given by

6.10 Modeling of Particle–Solid Interaction

π(ρmax )2 lp = n−1 p o .

107

(6.176)

Combining (6.174) with (6.176) leads to (6.169). The usage of realistic interaction potentials requires a considerable effort in evaluating the scattering angle α, since the conversion (6.30) should be done numerically, an effort which is not fully reduced by applying the so-called “magic” scheme invented by Biersack and Haggmark [266]. In the range of reduced energies 0.1 < ε < 10, the following approximation can be of use [279] sin2

α 2

=

1 1 + [C1 ε (ρp /aL )C2 ]2

(6.177)

with C1 = 4 − 2 log10 ε and C2 = 1.4 − 0.4 log10 ε. For energies ε > 10, the analytical relation derived for the Coulomb potential (see (6.32)) is valid sin2

α 2

=

1 1 = 1 + (2pEr /Ac )2 1 + (2ερp /aL )2

(6.178)

with the reduced impact parameter ρp /aL and Ac = Z1 Z2 e2 /(4πo ). The given algorithm (steps 1–7) can be readily translated into a computer language resulting in a code whose core routine consists of a few dozen lines only, but is able to produce results which are inaccessible to theoretical analysis, and that just by exploiting a common PC. The assumption of independent particle motion cannot be held in any case, since changes in composition and structure produced by one particle naturally affect the motion of the others. These highly non-linear effects are the subject of so-called dynamical simulations (for example see the description of the TRIDYN code in [280]). The Monte Carlo method is much faster than molecular dynamics simulations and runs with several million particles are not unusual. However, its restriction is closely connected to the used assumption of binary collisions, which is not valid in the sub-eV energy range, as discussed already in Sect. 5.4.

7 Electrical Coupling

Beginning from Langmuirs initial and fundamental study [281,282] in 1929 concerning the contact of a neutral plasma with an uncharged material surface, the related questions have been the subject of intense investigation. Up to now, especially by using the so-called Langmuir probes to measure plasma density and temperature in plasmas, some points are still open [283, 284]. Under steady state conditions, the electron flux and the ion flux to a floating plate immersed into a plasma must be equal. Frequently, the electrons in a plasma have a larger temperature than the ions, i.e., Te > Ti , but even for equal temperatures, the electrons, because of larger velocity, quickly charge a plate exposed in the plasma. A potential difference, called the Langmuir sheath potential, between the plate and the plasma is established, such that almost all electrons, except the fastest, are repelled, and the slower ions are attracted in order to satisfy the steady ambipolarity condition Γi = Γe . This sheath region is extended over only a few Debye lengths in the cases without magnetic field. The flux density  (7.1) Γ = f (v) v d3 v is determined by the velocity distribution f (v) of the particles. In thermalized plasmas with temperature T , a Maxwellian distribution (see Sect. A.1)
3

f (v)d v = n∞

m 2πkB T

(3/2)


 mv 2 exp − d3 v 2kB T

(7.2)

is established, where n∞ is the density of the particles in the undisturbed region. However, in the sheath the velocity distribution of the particles is strongly distorted due the fact that in this thin region the electrostatic forces are stronger than the inertial forces. The trajectories of charged particles near the surface differ significantly from those in the undisturbed plasma.

110

7 Electrical Coupling

7.1 Electron Flux Density In general, all electrodes immersed in a plasma are negatively charged with respect to the surrounding plasma, i.e., φelectrode < φplasma , irrespectively whether they act as an anode or cathode in the electric circuit. The reason for this is that the fast electrons (at the same temperature as the ions they have a much higher velocity due to their smaller mass) have to be repelled from the electrodes, otherwise the electron current out of the plasma would be too high and could not be compensated by an ion flux. A preferential loss of electrons would result, and the charge neutrality of the plasma and, therefore, the plasma itself would be destroyed. The flux density of electrons along the x-axis, which is directed away from the probe surface into the plasma, is given by
Γ (x) = n∞ ∞ × −∞

me 2πkB T

(3/2) ∞ −∞



me vy2 exp − 2kB Te

 dvy

−v

   min me vz2 me vx2 exp − vx exp − dvz dvx 2kB Te 2kB Te

(7.3)

−∞

taking (7.2) as the velocity distribution near the sheath entrance. Electrons with vx > 0 move away from the surface and cannot reach it. With a retarding potential, only electrons with sufficient velocity vmin in the direction toward the surface have the chance to reach the surface. Electrons in the velocity interval vx = [−vmin , 0] are pushed away. Because of energy conservation, the sum of potential and kinetic energy is constant along x E=

me vx2 me vx2 + Qφ(x) = − eφ(x) . 2 2

(7.4)

In the undisturbed plasma far away from the probe, the potential is usually set to zero, i.e., φ(x = ∞) = 0. An electron with a certain velocity vx=∞ = v∞ in the undisturbed plasma can only reach a certain position x if the condition     E = E ∞ x me vx2  E∞ ≥ − eφ(x) (7.5)  2 vx =0 2 is satisfied. In the point of return, vx = 0. E∞ = me v∞ /2 should be at least equal to −eφ(x) (note that φ(x) < 0). Hence, the minimum velocity required to reach a position x yields  −2eφ(x) vmin = . (7.6) me

7.2 Ion Flux Density

111

Note that vmin is always positive. The minus sign in the limit of the integral in (7.3) indicates that negative velocities in the x-direction are required to approach the probe surface due to the definition of the x-axis. Performing the integrations in (7.3) yields √

−2eφ(x)/me
  me me vx2 vx exp − dvx Γe (x) = n∞ 2πkB Te 2kB Te −∞ 
 kB Te −eφ(x) = −n∞ exp − 2πme kB Te



−

(7.7)

or, as given in the familiar form with the mean electron velocity ve  = 8kB Te /(πme )
 eφ(x) n∞ ve  Γe (x) = − exp . (7.8) 4 kB Te The integrals over vy and vz give each 2πkB Te /me . For φ(x) = φelectrode , i.e., x = 0, the relation (7.8) yields the electron flux reaching the electrode. The prefactor in (7.8) is equal to the Boltzmann factor appearing by analyzing the transformation of the velocity distribution in a potential field. As known, the flux of particles with a Maxwellian velocity distribution through a arbitrary surface is n∞ v/4 (Sect. A.1).

7.2 Ion Flux Density The analysis is considerably simplified if the ions are assumed to be cold, i.e., Ti = 0. In fact, the exact results differ only marginally from the obtained results using the cold ion model [285]. This is due to the large ratio of the ion mass to the electron mass. Even at Ti  Te , the velocity ratio (and therefore the flux ratio, since ne = ni in the undisturbed plasma) remains almost unchanged. With Ti = 0, the ion velocity away from the surface is zero, but for Ti  Te the ion velocity is still much smaller than the electron velocity. To achieve balance of fluxes, the ions have to be accelerated toward to surface. In the attracting potential of the so-called presheath, they obtain a certain velocity. The velocity at the sheath entrance vs is determined by energy conservation mi vs2 0 = Es + Q φs = + Q φs (7.9) 2 leading to  2Q φs (7.10) vs = − mi where Q is the ion charge and φs the potential drop between x = x∞ and the position x = xs at the sheath entrance. For a position x in the sheath

112

7 Electrical Coupling

(0 ≤ x ≤ xs ), energy and flux conservation give Es =

mi vx2 + Q φ(x) 2

(7.11)

and ns vs = nx vx

(7.12)

since no sources or sinks are available inside the sheath. Combining (7.11) and (7.12) leads to an expression for the ion density in the sheath   vs 2Es 1 Es . (7.13) n(x) = ns = ns = ns vx mi v x Es − Q φ(x) The ion flux density Γi = −ns vs

(7.14)

is determined by the density at the sheath entrance ns and the ion velocity there. Since we have not yet determined the potential φs , the value of vs remains so far unknown. In the next section, a new derivation is presented. It is based on a criterion of minimum energy. Considering a source free sheath and presheath, the question comes up, how the continuous loss of charged particles at the surface is compensated, since the plasma otherwise would be rapidly extinguished when a material surface is immersed into it. The ions are recombining with surface electrons and leave the surface either as fast backscattered neutrals or slow emitted molecules. Most of the incident electrons contribute to the surface charge of the material. In a first approximation, the material surface can be regarded as fully absorbing. In equilibrium, a plasma source is clearly required. However, the incorporation of a source into the theoretical models is difficult. Besides the transition from the presheath to the sheath, a new transition border separating the plasma source region from the presheath is then needed with the problem of defining appropriate boundary conditions. Usually, in order to simplify the analysis, a flux according to a half-Maxwellian distribution is assumed to be injected into the presheath. While the dimension of the electric sheath is more or less well-defined (a few Debye lengths), the dimension of the presheath is still an open question. As described in the next section, the potential drop in the presheath is small enough to be sustained in the plasma on a much larger spatial scale length than the Debye length. In the experiments, the presheath region overlaps into the plasma source region. A clear distinction—as a theorist would desire to make—is not possible. In magnetized plasmas, most of the plasma particles are supplied into a certain flux tube by transport across the magnetic field lines. To formulate and to analyze a consistent model is a challenging task.

7.3 Bohm Criterion with the “=” Sign Two regions can be distinguished: (1) the electric sheath, a thin layer with the thickness of a few Debye lengths where charge neutrality is violated, and (2)

Sheath

7.3 Bohm Criterion with the “=” Sign Plasma Te i e >0, Ti=0

Potential

113

φ=0 φs

Presheath φw

Fig. 7.1. Potential distribution in the plasma disturbed by an immersed material probe

the presheath, an extended region where quasineutrality prevails (Fig. 7.1). The potential at infinity, i.e., in the undisturbed region, is here set to zero (φ∞ = 0). The term φs denotes the potential at the sheath entrance and gives the potential drop in the presheath and φw is the potential at the wall (Fig. 7.1). Let us describe the ion flux toward the plate simply by (7.14) Γi,s = npl,s vi,s

(7.15)

where vi,s is the ion velocity and npl,s is the plasma density at the sheath entrance. In the undisturbed plasma region, the ions usually do not have a notable flow velocity. In the small presheath potential drop, however, they are accelerated up to vi,s at the sheath edge. In addition to the assumption Ti = 0, it is assumed that the ions starting with vi,∞ = 0 far away from the probe and fall then collisionlessly through the presheath potential drop. Using the equation of energy conservation gives (7.10)  −2eφs (7.16) vi,s = mi for singly charged ions (Q = +e). The electron flux in the sheath is given by the relation (7.8), which serves as an excellent approximation  # e(φw − φs ) npl,s ve  exp . (7.17) Γe,w = 4 kB Te Note that the fluxes of electrons and ions are spatially constant in the sheath. They are equal at each position. While the ions are simply accelerated toward the plate (their density is decreasing but their velocity increases), the electron flux consists of energetic electrons moving toward the plate and reaching it, and of less energetic electrons, which are repelled and turned back from the collector. Summarizing both contributions results in an electron flux directed toward the plate equal to the ion flux and that at each position in the sheath. Substituting the relations (7.15) and (7.17) together with (7.16) into the ambipolarity condition Γi,w = Γe,w yields, after some rearrangements,
 4vi,s kB Te φw = φ s + ln e ve  

−eφs 4πme kB Te = φs + + ln . (7.18) ln 2e mi kB Te

114

7 Electrical Coupling

Fig. 7.2. The wall potential as a function of the presheath potential drop according to (7.18) with me /mi = 1/1836. The wall potential φw attains a minimum for φs = −(0.5 kB Te /e)

It is worth noting that for a non-conducting surface the balance of electron and ion fluxes is established locally at each point of the surface by the arising potential in the electric sheath according to the electron temperature right above this position. In the case of a conductor, this criterion is replaced by  (7.19) 0 = (Γi − Γe ) dS where the integration has to be performed over the whole surface of the conductor, and the plasma parameters as well as the fluxes may differ with position. For given Te , mi , and me , there is equation (7.18), but with two unknown potentials φw and φs . As the required additional relation, a minimum energy argument may be applied. Plotting φw (7.18) as a function of φs (see Fig. 7.2) we notice that there is a minimum. It is easily found by differentiating (7.18) dφw 1 kB Te =0=1+ dφs 2 eφs

(7.20)

leading to kB Te . 2e Inserting (7.21) back into (7.18) defines the wall potential 


2πme kB Te φw = −1 ln 2e mi φs = −

(7.21)

(7.22)

including the potential drop (kB Te /2e) ln[2πme /mi ] in the sheath [281] and the potential difference in the presheath of −kB Te /(2e). The potential φw given in (7.22) is called floating potential, since no currents are drawn from the collector by an external circuit.

7.3 Bohm Criterion with the “=” Sign

115

Not only does the potential at the wall φw but also the energy flux density arriving the surface  m v 2 vf (v, φ)d3 v (7.23) P (φ) = 2 has a minimum for the wall potential given by (7.22). This is not surprising, due to the fact that besides thermal contribution, the energy flux is linked directly to the value of the wall potential as the potential difference between the surface and the undisturbed plasma region. As seen in (7.23), the particle distribution function depends on the potential. Hence, the principle of minimum energy flux could also have served as a criterion to derive (7.21). It was shown in experiments [286] that the energy flux reaches a minimum when attaining the floating potential (if ion reflection is not considered, i.e., RE = 0). With the potential drop in the presheath (7.21), the ion velocity at the sheath entrance is given by the ion sound speed cs (for Ti = 0)  kB Te vi,s = = cs (7.24) mi according to (7.16). This is, in fact, the Bohm criterion, but obtained here with the equal sign in contrast to other derivations stating vi,s ≥ kB Te /mi . The question, whether the equals sign holds or not, led to an extended discussion in the literature, see for example the review articles [287–291]. Most of the complications arise from the more or less artificial division of the plasma into sheath and presheath regions with the problem of defining correct boundary conditions. The ion flux density toward the surface is then given by Γi = ns cs .

(7.25)

Since the plasma is recombining at an exposed material surface, a source to replace the charged particles as well as a energy source is needed. The energy required for building-up the potential difference between the plasma and the material is delivered by the hot electrons and is supplied to ensure stationary conditions. In the case of a very localized plasma source, the presheath potential drop occurs just close to its location [285, 292]. If the plasma is generated by ionization and heated in a larger volume or is provided by cross-field transport, the presheath will be distributed over a larger region. The necessity to describe the whole system including the plasma source, the presheath, considering also collisions, and the sheath renders analytical solutions impossible and code simulations are required. The most powerful simulation tool for analyzing the plasma–material transition is the so-called PIC (particle-in-cell) method (see Sect. 7.6). Considering the electric sheath, parameters such as the reflection and emission coefficients, ion temperature, collisionality, magnetic field geometry, and the impurity concentration are of importance.

116

7 Electrical Coupling

With increasing ion temperature ratio τ = Ti /Te , the potential in the sheath and in the presheath drops, since less effort is required to accelerate the ions up to the sound speed at the sheath entrance. However, the affected change of the potential distribution is rather small [285, 292] and could be related to the remaining large discrepancy between electron and ion velocities even for Ti ≥ Te . Riemann has shown [293] that the Bohm criterion is also satisfied in a collisional plasma, but the ion acceleration in the presheath is affected by collisional friction. The potential drop across the collisional zone can be larger in magnitude than the presheath voltage drop for a plasma with no collisions. The additional potential drop is required to accelerate the ions to the sound speed in the presence of ion momentum loss. With higher collisionality, the potential drop increases modestly with a logarithmic dependence. Processes such as ion reflection and sputtering do not have a direct effect on the electric sheath, since incoming ions recombine with surface electrons and are reflected as neutrals; as well, almost all sputtered particles are neutral atoms. This is in contrast to electron emission. Each emitted electron which moves away from the surface is equivalent to a plasma ion moving toward the surface considering the current balance. In the (improbable) case of insheath ionization of reflected and emitted atoms, new charge carriers would appear and a plasma source would be established near the surface resulting in a change of the potential distribution [294].

7.4 Space Charge Limited Currents Under certain conditions, a region of negative space charge is established in front of the surface and causes a limitation of electron emission, for example, of thermionic emission of electrons [295–298]. More accurately, the electron current into the plasma is limited, not the emission current out of the surface which, in the case of thermionic emission, is a function of the surface temperature only. A large part of emitted electrons, however, is immediately forced back to the surface by the electric field in that region of negative space charge. This effect starts to play a role when the electric field at the surface becomes zero. The ion current density ji and ion density ni as well as je and ne for the plasma electrons are given in the cold ion approximation (Ti = 0) by the expressions [295, 299] (see also (7.15,7.17))   2Ei Ei ji = enpl , ni = npl mi Ei − eφ " !
 eφw ve  jce /e exp je = −e npl − 4 kB Te 2(−eφw )/me

7.4 Space Charge Limited Currents

! ne = npl −

jce /e 2(−eφw )/me

"


exp

eφ kB Te

117

 (7.26)

where Ei denotes the ion energy at the sheath entrance. In the derivation of (7.26), the neutrality condition npl = ne + nce = ni is applied, where npl is the plasma density in the undisturbed region. The current density of the electron emission jce is given by the Richardson–Dushman relation (6.154) in the case of thermionic electron emission, and their density in the sheath is defined by nce =

jce jce /e = evce 2(eφ − eφw )/me

(7.27)

where vce is the velocity of the emitted electrons which fall freely through the sheath. The derivation of the expression (7.27) is similar to that of the ion density (7.13) in Sect. 7.2. The potential at the wall φw as deduced from the current balance ji + je + jce = 0 !  2Ei + jce /(enpl ) eφw = kB Te ln mi
T
 ve jce me /(2(−eφc )) . (7.28) 1− 4 enpl The ion energy at the sheath entrance Ei can be determined using the Poisson equation d2 φ e = (ne + nce − ni ) (7.29) dx2 o from which a relation for the electric field E is easily obtained (multiplying by dφ/dx and integrating from ∞ to x) 1 2 1 E = 2 2




dφ dx

2

φ =

e (ne + nce − ni ) dφ ≥ 0. o

(7.30)

0

Using in (7.30) the expressions of (7.26) for the densities and expanding eφ/Te and eφ/Ei in a Taylor series yields a limit for the ion energy at the sheath entrance (φ(x = ∞) = 0) 

kB Te Ei jce me kB Te + (−eφw )−3/2 − eφw . Ei ≥ (7.31) 2 npl e 2 2 For zero emission (jce = 0), this relation reduces to the Bohm criterion, i.e., Ei ≥ kB Te /2 (Sect. 7.3).

118

7 Electrical Coupling

Finally, postulating zero electric field at the cathode, a condition for critical ∗ can be defined by integrating (7.30) emission current density jce




∗  eφw me jce −1 0 = 1− kTe exp npl e 2(−eφw ) kB Te " !  ∗ 2jce me eφw − (−eφw ) + 2Ei 1− −1 . (7.32) npl e 2 Ei ∗ , even in the The emission current cannot exceed its critical value, jce ≤ jce case of increasing surface temperature.

7.5 Effect of Magnetic Field Geometry Under oblique incidence, two parameters determine the behavior of the sheath: (1) the “magnetization” parameter ξ (11.2) that compares the gyro-radius ρ of electrons and ions with the Debye length λD , and (2) the angle α between the magnetic field lines and the surface plane. In fusion relevant experiments, the electron gyro-radius is nearly equal to the Debye length, i.e., ξe  1, but the ion gyro-radius is much larger, i.e., ξi  1. In order to reduce the heat load onto the divertor and limiter plates, the angle α is kept as small as possible, usually about 1 to 3o . Then, the presheath reveals a double structure. It is composed of a collisional presheath and a magnetic presheath. The thickness of the magnetic presheath is found to be approximately (cs /ωi ) cos α and varies with ion mass, electron temperature, magnetic field, and angle α [300–302]. For α < 3o , the thickness is approximately equal to the ion gyro-radius ρi  cs /ωi . The collisional presheath thicknesses vary with the ion-neutral collision mean free path and/or the extent of the plasma source region (Fig. 7.3). As in the case without a magnetic field, the presheath has the function to accelerate the ion to sound speed. This value is reached at the entrance of the magnetic presheath along the magnetic field lines. In the magnetic presheath,

Plasma Presheath

Magnetic Field Line Ion

Electric Field

Magnetic Sheath

Electron

Electrostatic Sheath

Surface

Fig. 7.3. Schematic of the different plasma zones in the near-surface region

7.5 Effect of Magnetic Field Geometry

119

the ion trajectories are bent over in a way that at the entrance of the electric sheath the sound speed is already reached with respect to the surface normal. The electrostatic sheath with a thickness of about the Debye length is characterized by the break-down of quasineutrality, i.e., by the onset of space charge effects, causing strong electric fields, whereas quasineutrality still holds in the collisional as well as in the magnetic presheath. The potential difference (see (7.22) and (7.28)) between the surface and the undisturbed plasma remains nearly the same regardless of the angle of incidence α [300]. However, at oblique incidence, the floating potential decreases and may become positive with respect to the plasma potential as shown in experiments in the plasma generator PSI-2 [303, 304]. In the cold ion model, the ion flux density is equal to Γi = npl vi sin α and the electron flux density is about Γe = (1/4)npl veT exp(eφw /kB Te ) sin α, since the electrons are bounded to the magnetic field lines up to the entrance of the electric sheath. So that the flux balance Γe = Γi , results in a potential φw are independent of α. These results hold as long as the angle is larger than one degree. In the case of perfect parallel magnetic field lines (α = 0), which in practice can hardly be achieved, the surface potential becomes positive, i.e., φw > 0 [305], since the ions with their larger gyro-radius should be repelled and the electrons must be attracted. For steady state, an ambipolar flux of ions and electrons toward the surface should be provided by collisions and/or anomalous transport. The velocity distribution of ions, which are accelerated in the sheath, is distorted in a way that the average impact angle with respect to the surface normal is about 60 degrees [306, 307]. Sputtering of the surface by the ions is enhanced, owing not only to the ion energy increase but also due to the shallow angle of impact according to the angular dependence of physical sputtering (6.119). Grazing incidence of the magnetic field lines affects also the effective emission of electrons. Some of emitted electrons might be led back to the surface by gyration. The crucial parameter here is the energy of the emitted electrons, which is, for example, different in the processes of thermionic emission and secondary electron emission. The latter process is characterized by emission energies of a few eV up to some tens of eV, while hot surfaces emit electrons with an energy corresponding to the surface temperature of about 0.1–0.2 eV. Electrons with small velocities have a larger probability to escape from the surface. An electron with nearly zero energy might be returned after one gyration, but its velocity component with respect to the surface normal will be as small as it was at the time of emission, while during the gyration the electron increases its velocity along the magnetic field lines owing to the electric field of the sheath. The Lorentz force has the tendency to return the electron to the wall while the electric field force pushes it away from the surface. If an electron is not absorbed during the first gyration, it will propagate into the plasma. With increasing energy of the emitted electrons and decreasing angle of the magnetic field lines with respect to the surface, the escape probability

120

7 Electrical Coupling

becomes smaller and, correspondingly, the effective emission into plasma is reduced [308]. In the case of magnetic field lines which are directed almost parallel to the surface, its roughness starts to affect the escape probability strongly, since for real materials the roughness is usually larger than the gyroradius of the emitted electrons.

7.6 Modeling of the Electric Sheath An appropriate description of the electric sheath is not possible without a full kinetic treatment. Fluid approximations or even gyro-kinetic methods fail at considering the transformation of the velocity distribution in the sheath. To reduce the effort of MD calculations, which deals with N ×(N −1) interactions in a system consisting of N particles, an averaged field can be introduced. This field is established by all the particles in the system. The transition from the description in terms of forces between the particles to a model in terms of a force field has been already mentioned in Sect. 5.7. A single particle moves in the field of other particles. Using the Poisson equation the potential distribution and, subsequently, the electric field E = −∇φ can be determined just by knowing the positions of all charged particles at a certain time. The accuracy of the field description depends only on the spatial resolution, i.e., the cell size of the applied numerical mesh. For example, a spatial resolution on the order of the interparticle distance would result in a high accuracy but would mean a higher computational cost. The particles are assigned according to their positions to grid points. By the way, this operation removes the uncertainty in describing the infinite potential at zero distance from the particle. The cell size should not be smaller than the Debye length (Sect. 5.2), otherwise the plasma cannot be correctly modeled. Such an inadequate choice would be promptly penalized by a diverging, strongly fluctuating numerical solution. On the other hand, collisions with an impact parameter smaller than the cell size are neglected (see the discussion in Sect. 5.4). Fluctuations of the potential at the grid points are alleviated by increasing the number of particles in one cell. In such simulations, the number of used particles are much smaller than the particle number in real plasma systems. The particles in the simulation act as representatives of a group of real particles having their averaged properties—not the cumulative ones. Comprehensive reviews on the so-called particle-in-cell (PIC) methods together with a detailed description of the required numerical methods are given in [309, 310]. 7.6.1 Principles of PIC Simulations The motion of “representative” particles, ions, and electrons are followed in time, while at each time step the density distribution of the particles as well as the corresponding potential distribution are evaluated on a mesh. Different

7.6 Modeling of the Electric Sheath

121

interpolation techniques can be applied to assign the particle position to the grid points and to obtain the field values for a certain position. The magnetic field B can be assumed in many applications to be fully determined by an external, and constant magnetic field Bo . This assumption is valid as long as the internal currents are sufficiently small to generate significant internal magnetic fields. A PIC simulation can be structured as follows: 1.

2.

3.

Definition of the initial conditions. At time t = 0, the position and velocities of all particles (ions and electrons), and thus the particle distribution function f (v, r, t = 0), has to be defined. Definition of time step ∆t, cell size ∆x and the number of particles N max . The numerical model is scaled to the real system using the factor npl Vpl (7.33) fnorm = max N where Vpl is the considered (real) plasma volume and npl the plasma density. The particles are assigned to the grid points to obtain the density distribution. Two of the most common methods of interpolation are shown in Fig. 7.4 and Fig. 7.5. The NGP method assigns the particles simply to the nearest grid points. The density at a grid point i is then given by ni = Ni /∆x where Ni is the number of particles at the grid point i. Using the CIC method (cloud-incell), both grid points in the neighborhood receive a “fraction” of the particle charge Qc

∆x − (x − Xi ) x − Xi Qi = Qc and Qi+1 = Qc (7.34) ∆x ∆x ∆x Xi

x

x

Xi-1

Xi+1 X i + ∆x 2

Xi - ∆x 2

n(x)

x

Xi ∆x

Fig. 7.4. NGP method

122

7 Electrical Coupling

Fig. 7.5. CIC method

5.

realizing a linear interpolation scheme. Xi , Xi−1 , and Xi+1 denote the grid point positions. The distribution of the potential and the electric field on the grid points are obtained by numerical solution of the Poisson equation satisfying the given boundary conditions. The value of the electric field at a certain position x is given by the value of the nearest grid point (NGP method) or by the values at the two grid points in the neighborhood

x − Xi Xi+1 − x (7.35) Ei + Ei+1 , E(x) = ∆x ∆x

6.

if Xi < x < Xi+1 , according to the CIC method. All particles are moved forward by one time step to new positions according to the equation of motion

4.

dve,i = F = Qe,i (E + [v × B o ]) . (7.36) dt The numerical solution of (7.36) is given in Sect. A.2. Steps 3–6 are repeated, until the desired calculation time tmax = Nt ∆t is reached, where Nt is the number of time steps. Evaluation and output of physical quantities and distributions. me,i

7. 8.

It is very important to use the same interpolation scheme for the density (step 3) and force (step 5) calculations, otherwise non-physical results such as the motion of a particle in its own field may occur. The extrapolation of the presented algorithm to two and three dimensions is straightforward. Usually, the equation of motion is in any case calculated in three dimensions, but a fast solver of the higher-dimensional Poisson equation should be chosen and the interpolation schemes should be extended to consider 4 neighborly grid

7.6 Modeling of the Electric Sheath

123

points in the case of two-dimensional simulations and 8 grid points in the case of three-dimensional simulations. 7.6.2 Boundary Conditions Due to the fact that in the simulation a certain plasma region is modeled, the definition of boundary conditions for the particles and the electric potential is essential. Numerical instabilities arise, for example, if a particle which has passed one border is put back into the simulation region at a position with a higher potential. This may happen if periodic boundary conditions for the particle but aperiodic conditions for the potential are defined. A periodic region is usually thought of as a fraction of an infinite extended plasma. Integrating the Poisson equation dE/dx = ρ/o with the charge density ρ = Qi ni − e ne over the system length L x+L 

x

1 dE dx = E(x + L) − E(L) = dx o

x+L 

ρ dx = x

L ρ o

(7.37)

shows that the total charge of the system should be zero in the case of periodic boundary conditions, E(x+L) = E(L). It is thus contradictory to demand periodic boundary conditions without ensuring zero total charge in the modeled system. In addition, the average electric field E in a region with periodic boundary conditions must also be zero since x+L 

∂φ dx = φ(x + L) − φ(L) = 0 = − ∂x

x

x+L 

E dx = −L E.

(7.38)

x

If a particle has left the region at one boundary, it is put into the system at the opposite boundary with the same velocity (in value and sign) in the case of periodic boundary conditions. Particles which belong to the two half-intervals [0, ∆x/2] and [L − ∆x/2, L] at the borders are counted up and assigned to the grid points at x = 0 and x = L to ensure equal density there. Having one or more material surfaces in the plasma, the condition of total zero charge must be given up, since at the surfaces charges can be collected. The resulting electric fields caused by a surface charge density σ are given by Eo =

σo o

and

EL = −

σL o

(7.39)

if one surface is located at x = 0 and the other at x = L. A thin plasma layer of thickness ∆x and charge density ρ can always be considered as an electrode with surface charge density of σ = ρ∆x. Particles (ion or electron) which strike the surface of electrodes are taken away from the list of “active” plasma particles and contribute to the corresponding surface charge. This models recombination at the surface. The

124

7 Electrical Coupling

simplest way to establish an equilibrium situation, i.e., particle conservation, is to feed an ion–electron pair into the plasma region when an electron and an ion are lost at the electrodes. In the Poisson equation, densities have to be calculated. For the grid points at the electrodes, we have, for example, using the NGP scheme,  (all particles) σo [0,∆x/2] + (7.40) ρ(0) = ∆x (∆x/2) Spl 

and σL + ρ(L) = ∆x

(all particles)

[L−∆x/2,L]

(∆x/2) Spl

(7.41)

where Spl is the plasma cross-section. 7.6.3 Choice of Time Step and Spatial Resolution Whether a simulation is successful depends on the right choice of three parameters: (1) the cell size ∆x for the calculation of the density distributions and the Poisson equation, (2) the time step ∆t by following the motion of the particles, and (3) the number of particles N max used in the simulation. Due to numerical approximations, error fields can hardly be avoided. These non-physical electric fields lead to an increase of the kinetic energy of the particle in time—a process called stochastic heating. The electrons with their smaller mass are especially affected. Better interpolation schemes (preferring the CIC method to the NGP scheme) and smaller time steps help to reduce the problem. Fluctuating error fields affect the trajectories of the particles leading to a collision-like behavior. By increasing the number of particles in each cell, this effect of “numerical” collisions can be minimized. The number of particles should be at least N max λD /l = 20, much better is a value of N max = 200

l λD

(7.42)

where l is the length of the considered plasma system. Only if the Debye sphere is properly filled with particles, can the occurrence of potential fluctuations be restricted. The grid size should be smaller than the Debye length ∆x < λD

→

∆x ≈ 0.2λD .

(7.43)

Usually 5 grid points are sufficient to resolve the Debye length. The time step has to be chosen in accordance to the fastest process—plasma oscillation. The gyration of electrons is also characterized by a frequency of the same order of magnitude in many plasma experiments, i.e., ωce  ωpe (see Sect. 11). Such a time step ∆t 2π/ωce is much smaller than actually required for the much

7.6 Modeling of the Electric Sheath

125

slower ions. Many simulations, especially in the past, have been performed by using another mass ratio mi /me instead of the correct one with the aim to use larger time steps. A ratio of mi /me = 100 is not unusual. This approximation is justified, since many processes show an asymptotic behavior with increasing mass ratio. The Courant–Friedrichs–Lewy criterion [311] connects the time max of the particles, which step and the grid size to the maximum velocity ve,i can occur in the simulation max 1 > (ve,i ∆t)/∆x > 0.1 .

(7.44)

No particle should be able to leap over one grid cell within one time step. On the other hand, it makes no sense to keep one particle too long in one cell, since the charge density in that cell remains then unchanged. Given a system length l, a plasma density npl , the masses me and mi , and temperatures Te and Ti the parameters of the numerical simulation can be chosen as  ∆x = 0.2 o kB Te kB Ti /[npl e2 (kB Te + kB Ti )] ∆t = 0.5

∆x kB Te /me

N max = 200

l o kB Te kB Ti /[npl e2 (kB Te + kB Ti )]

(7.45)

since ∆t = 0.1/ωce  0.1/ωpl = 0.1λD /ve = 0.5∆x/ve . Note that under the conditions of today’s fusion experiments, λD ≈ ρe and ωpl ≈ ωce . To get an idea of the computational costs, the repetition number N cycle of the main cycle (steps 3–6 in Sect. 7.6.1) can be estimated. Let the calculation region be 10 times larger than the ion gyro-radius, i.e., l = 10 ρci , and the simulation last 10 times the ion gyration time, i.e., tmax = 10 Tci . Then, the number of repetition is about tmax N cycle = Nt · 2 · N max = 2 N max ∆t 
 10Tci kB Te /me 200 · 10ρci cycle N = 2 0.1λD λD 
  10 · 2π mi kB Te /me 200 · 10ρci 2 N cycle = 0.1λD eB λD
 3/2 mi ρce ρci m ∝ √ i npl . N cycle  2 × 106 2 me λD me

(7.46)

The factor of 2 appears because two species (positively and negatively-charged particle) are involved. Essential are the mass ratio and the density. For a (pure academic) mass ratio of mi /me = 1 and the set of parameters: l = 0.02 m,

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7 Electrical Coupling

Te = Ti = 10 eV, npl = 1014 1/m3 , mi = me = 1 amu, Qi = e, Qe = −e, and B = 0.2 T, we have ∆x = 3.3 × 10−4 m, ∆t = 5.3 × 10−9 s, N max = 200 · l/λD  2400, and N cycl = 4 × 106 . On a common PC, this takes not longer than one minute having a CPU time of roughly 10−5 s for one cycle. For more realistic parameters, i.e., mi /me = 1836 and npl = 1020 1/m3 , the effort increases by five orders of magnitude. Such simulations are very time consuming and often simplifying approximations have to be applied, even when using supercomputer systems. Note that an extension of the model to two or three dimensions is readily programmable, but the effort increases with the power of the dimension number.

8 Power Coupling

The electric sheath established via the contact of a plasma with a material surface acts as an energy transfer zone. While the electron and ion energy fluxes vary with distance to the surface, the sum of both contributions remains independent of the position above the surface, since there are no sources or sinks of energy in the sheath. The ions gain their energy from the electrons by being accelerated in the electric field, which is generated at the expense of the electron kinetic energy. The transfer of the electron kinetic energy (out of the tail of their Maxwellian velocity distribution) via the potential energy of the electric field in the sheath to the kinetic energy of the ions can be adequately described only by numerical simulations, as detailed in Sect. 7.6. The heat flux through the sheath (7.23) is determined by the particle distribution function, which is strongly distorted in the sheath, and the concept of temperature becomes, therefore, questionable. Nevertheless, the simplified analytical relations given in the following sections help to understand the main effects and dependencies, even though the transformation of the velocity distributions cannot be addressed there.

8.1 Heat Flux Densities The heat flux density from the plasma to the wall Pw = Pi + Pe is given by the contribution of ions

 ion ion atom (1 − RE )(2kB Ti + |eφw |) + Iion − W − Eex Pi = Γ i R N

ion +Γi (1 − RN ) 2kB Ti + |eφw | + Iion  molec (8.1) −W + (Ediss − Eex )/2 − Etherm and of electrons ele ele ele 2kB Te (1 − RE ) + Γe (1 − RN )(2kB Te + W ) Pe = Γe R N SEE −Γe δESEE (EESEE + W )fesc SEE em −Γi δISEE (EISEE + W )fesc − Γem Eem fesc

(8.2)

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8 Power Coupling

with the ion and electron flux density according to (7.25) and (7.8)  # npl e(φw − φs ) ve  exp sin α Γi = npl cs sin α , Γe = 4 kB Te

(8.3)

at the sheath entrance, respectively. The assumption of a sine dependence of the fluxes remains valid down to grazing angles of magnetic field line incidence, i.e., for α ≥ 1o [312]. The density at the sheath edge is simply set equal to the plasma density here. ion deposit only Those ions which are reflected with a probability of RN ion the part (1 − RE ) of their thermal energy flux density of Γi 2kB Ti (here a Maxwellian distribution of incoming ions has been assumed) and of the energy gain in the sheath |eφw | at the target. Equation (8.1) is valid for singly charged ions. A higher charge results in minor changes of the corresponding terms in (8.1). The ions recombine at the surface and release the ionization energy Iion minus the work function W , since the recombination process is accompanied by electron extraction from the bulk material. Further, energy (Ecx ) can be lost due to excitation of the reflected particles. Especially at low plasma temperatures, the recombination term in (8.1) is important and cannot be neglected [313]. Those ions which are implanted deposit their whole energy and are predominantly released as thermal molecules with an energy of Etherm , which corresponds to the surface temperature. In addition, half of the dissociation energy Ediss and excitation energy of the emitted molecule must be added. Electrons which are not reflected release the work function W (see (8.2)), because of their transfer from a free to a bounded state in the bulk material. Emission of secondary electrons can be caused by ion impact (ISEE) and electron impact (ESEE). The last term in (8.3) corresponds to thermionic SEE emission, where Eem is the energy of emitted electrons. The coefficients fesc em and fesc describe the escape probability of emitted electrons in the case of oblique magnetic field (see Sect. 7.5). As a rough estimation, the approximation fesc  sin α can be used. Often, a so-called energy transmission coefficient γE = Pw /(Γi kB Ti ) is introduced. Neglecting electron emission, γE becomes simply γE = 2 +

|eφw | 2kB Te Iion + W Iion + W + +  7.5 + kB Ti kB Ti kB Ti kB Ti

(8.4)

with Γe = Γi and the further assumption of Ti = Te ; for hydrogen plasmas φw is approximately equal to 3.5kB Te as shown in (7.22). With electron emission, the energy transmission coefficient can reach values up to 25–30 [216,314,315] until space charge limitation sets in. In order to explain this sudden increase in heat load, without referring to bulky relations, a reduced set of equations (8.1, 8.2) with only secondary electron emission by electrons should be considered here. Then, γE = (Pe + Pi )/(Γi kB Ti ) yields

8.2 Change of Surface Temperature

Γe (2kB Te − δESEE EESEE ) + Γi (2kB Ti + |eφw |) Γi kB Ti Γe = [2Te /Ti − δESEE EESEE /Ti ] + 2 + |eφw |/kB Ti Γi 1 = [2Te /Ti − δESEE EESEE /Ti ] + 2 + |eφw |/kB Ti 1 − δESEE

129

γE 

(8.5)

where the ratio Γe /Γi is obtained from the flux balance Γi = Γe − Γe δESEE = Γe (1 − δESEE ).

(8.6)

With increasing emission, the energy transmission coefficient increases too— due to the term 1/(1 − δESEE ) in (8.5). In other words, each emitted electron is directed away from surface and is, therefore, equivalent to an ion directed toward the surface with respect to the current balance. Hence, with higher electron emission, less ions are required of being attracted to the surface and the surface potential becomes smaller in absolute value, but of course remains still negative. With smaller potential difference over the sheath, more plasma electrons are streaming to the surface (see (8.3)) and increase the heat load. At higher electron emission (δESEE > 0.8–0.9), space charge limitation occurs (see Sect. 7.4) and the power transmission coefficient saturates at the high level of γE = 25–30.

8.2 Change of Surface Temperature The wall surface is cooled by heat conduction into the bulk of the wall, by thermal radiation, by emission of thermal electrons (already included in (8.2)), and by evaporation of the wall material. The density of heat flux conducted away from the surface Ps depends on the surface temperature Ts Ps = keff (Ts − Tbulk ) + εg σSB Ts4 + Γsubl Esubl

(8.7)

where keff is the effective coefficient of heat transmission in W/(m2 K), determined by the geometry and the heat conductivity, Tbulk is the temperature of the cooled side of the target, εg is the emissivity or grayness coefficient, σSB is the Stefan–Boltzmann constant, Esubl is the sublimation energy, and Γsubl is the flux density of the sublimated particle, which depends also strongly on the surface temperature Ts according to (6.127). The first term on the righthand side of (8.7) describes the energy losses due to heat conduction, and the second term the cooling by radiation. This highly non-linear equation can be solved by iteration in order to obtain the surface temperature in accordance with the heat flux balance of Ps = Pw . In general, the heat conduction equation cp ρ

∂T = div(k gradT ) + QE ∂t

(8.8)

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8 Power Coupling

should be solved together with adequate boundary conditions, which are usually highly non-linear according to the last two terms on the right-hand side of (8.7), to obtain the temperature distribution in the material. Neglecting the temperature and spatial dependence of the heat conductivity k in W/(m K) and having no inner heat sources, i.e., QE = 0, this equation reduces to ∂T /∂t = a∇2 T with the thermal diffusivity a = k/(cp ρ) in m2 /s, the material density ρ in kg/m3 , and the heat capacity cp in J/(kg K). The numerical solution of (8.8) can be obtained using the methods of finite differences or finite elements. In the case of actively cooled target structures of complex geometry, which consist of different materials and cooling media, the application of widely accepted commercial software packages such as ANSYS or FEMLAB is recommended. In special cases, an analytical description is possible, which can serve as a testbed for numerical analysis. Some of them are given in the next sections. 8.2.1 Heat Conduction in a Half-Infinite Medium In the case of a half-infinite space, the one-dimensional time-dependent solution of (8.8) is √
 
x x2 2P a t Px √ √ exp − (8.9) T (x, t) = To + − 1 − erf 4at k k π 2 at for a constant heat flux density onto the surface  ∂T  P = −k ∂x x=0

(8.10)

and an initial, temperature of the bulk material T = Tbulk at t = 0; x √ uniform erf(x) = (2/ π) 0 exp(−t2 )dt is the error function. According to (8.9), the √ rise of temperature at the surface is proportional to t and given by the often used relation T (x = 0, t) = Ts (t) = Tbulk +

√ 2 P t. πcp ρk

(8.11)

For many materials, the material parameter 2/(πcp ρk)1/2 is about 10−4 K m2 /(W s1/2 ). The relations (8.9) and (8.11) can be used to describe inertial cooling of a material layer, which is thicker than the dcrit  √ 2k dcrit = 2 a tmax = , (8.12) cp ρ tmax the depth reached by the thermal wave after a certain time tmax . As long as d > dcrit , the influence of the boundary condition at the x = d can be neglected.

8.2 Change of Surface Temperature

131

8.2.2 Point-like Heat Load If at point r o = (0, 0, 0) at the surface a certain amount of energy Wth is instantly released, then the corresponding temperature distribution is given by

y2 z2 x2 2 Wth − − exp − T (r, t) = (8.13) √ 4ax t 4ay t 4az t cp ρ (4π t)3/2 ax ay az where ax , ay , and az are the thermal diffusivities for the different directions. The loaded energy Wth is related to a heat flux density P

  W Wth [J] = P dS dt  P ∆t ∆S = P ∆t ∆y∆z (8.14) m2 ∆t ∆S

acting during a short time interval ∆t on a small surface area ∆S = ∆y∆z. In the case of a local, but constant heat load, the temperature distribution is obtained by integrating (8.13) over the exposure time 2 P ∆S T (r, t) = cp ρ (4π a)3/2

t 0



1 r2 exp − dt 4a(t − t ) (t − t )3/2

(8.15)

for a = ax = ay = az . With the substitution τ = 1/(t − t ), (8.15) is transformed into an integral of type  √ √ (8.16) exp(−c τ ) dτ / τ = π/c erf( c τ ) leading to T (r, t) = with r = have


r 2P ∆S 1 − erf √ ρ cp 4π a r 4at

(8.17)

x2 (y − yo )2 (z − zo )2 + + ax ay ay

(8.19)

x2 + y 2 + z 2 . Considering different diffusivities ax , ay and az , we
 r∗ 2P ∆S T (r∗ , t) = 1 − erf √ (8.18) √ ρ cp 4π r∗ ax ay az 4t 

with r∗ =

if the heat load is released at the surface element ∆S at the position r o = (0, yo , zo ). 8.2.3 Heat Conduction and Diffusion The equation of diffusion ∂n = div(D grad n) ∂t

(8.20)

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has the same mathematical form as the equation of heat conduction (8.8). The particle density (or concentration) corresponds to the temperature, the diffusion coefficient D to the thermal diffusivity a = k/(cp ρ). Releasing instantly at the point r = 0 a certain amount of energy Wth , the temperature distribution evolves in time as (see (8.13))

Wth r2 T (r, t) = exp − . (8.21) 4at cp ρ (4π a t)3/2 Analogously, the concentration profile of N particles, which start at r = 0 to diffuse, is described by

r2 N exp − n(r, t) = (8.22) 4Dt (4π D t)3/2 with the equivalence N≡

Wth [Km3 ] . cp ρ

(8.23)

8.3 Power Removal Limiter and divertor plates are especially subjected to high power flux densities (up to several tens of MW/m2 ) due to ion and electron bombardment and by radiation. For design purposes, typically 80% of the produced α power is assumed to be intercepted by the divertor (240–480 MW) leading to a load of 40–60 MW/m2 (see Sect. 9.1.5). Using additional means, such as radiation, this value should be reduced to 5–10 MW/m2 [5]. While below 1 MW/m2 passive (inertial) cooling is sufficient, active cooling is required for 1–10 MW/m2 . According to (8.11), the rise of temperature at the surface is proportional √ to t. The power flux density can be estimated for typical plasma parameters as (8.24) P = γE cs ne kB Te sin α  3.5 MW/m2 with ne = 2.5 × 1020 1/m3 , Te = Ti = 10 eV, mi = mD , α = 2o , and an energy sheath transmission coefficient of γE  8. For normal incidence of the magnetic field lines, i.e., α = 90o , a value of 100 MW/m2 would result. It is obvious from (8.11) that with P > 1 MW/m2 a surface temperature close to the melting temperature is reached after some tens of seconds. Hence, active cooling is quite essential regarding steady state operation. The heat removal capacity is limited by the thermal conductivity of the material. Since the heat flux density P through a layer of thickness d is constant in steady state, the integration of P = −k(T ) dT /dx =const. yields d

T0 k(T ) dT → Pmax

P dx = 0

Td

1 = d

T0max

k(T ) dT =

Λ(T0max ) − Λ(Td ) d

Td

(8.25)

8.4 Thermal Stress

133

using the integral thermal conductivity Λ(T ), which includes the temperature dependence of the thermal conductivity k(T ). Td is the temperature at the cooling side of the target plates. The temperature at the plasma-facing surface T0 is limited to the temperature at which melting (metals) or strong sublimation (in the case of graphite) starts. The relation in (8.25) determines the power flux density, which can be removed in steady state by one-dimensional thermal conduction through a material of thickness d for a given temperature difference. The choice of the thickness is always a compromise between a safety margin with respect to thinning by erosion and still tolerable cooling properties. Comparing different materials and alloys, it was shown in [215,316] that the maximum power flux density, which can be carried off by conduction is 5 to 20 MW/m2 . Using carbon-based materials such as CFC (carbon fiber composites) with their superior heat conductivity, much higher values can be achieved. However, at high radiation fluences degradation of the thermal properties of these materials occurs. Recently, robust technical solutions have been developed to handle steady state power flux densities up to 10 MW/m2 . Higher values can be tolerated only transiently and have to be avoided by aiming for plasma scenarios with tolerable plasma parameters near the surfaces.

8.4 Thermal Stress Another critical subject is thermal stress built up due to temperature gradients in the material in a certain geometry. The thermal stress σts (as force per unit area), which develops if a structure is completely constrained (not allowed to move at all), is the product of the coefficient of linear expansion αT , the temperature difference ∆T , and Young’s modulus EY for the material, i.e., σts = αT ∆T EY . During heating, the arising stresses are compressive in the near surface region and tensile in the bulk region. A thermal stress criterion can be derived by expressing the heat flux density P simply by P = k|∆T |/d [317] σ αT EY P d γσ = = 1), then a part of the surface will be plastically deformed by thermal cycling. In addition, this deformation leads to residual tensile stresses in the surface region during the cooling phase after plasma exposure. Almost all technical designs of target plates are based on the use of different materials, where additional stresses are induced owing to different thermal expansion properties. These stresses have to be kept well below the yield

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strength under all plasma conditions, i.e., also in the case of transient effects such as ELMs and disruptions. Extensive thermal testing of components subjected to repeated thermal loading is required for reliable lifetime predictions. In the case of thin films, for example, having a tungsten layer on graphite, the initial stress distribution built up during the deposition process is also of importance. There are thermal and intrinsic stresses. Thermal stress occurs because the films are usually deposited above room temperature. Upon cooling from the deposition temperature to room temperature, the difference in the thermal expansion coefficients of the substrate and the film cause thermal stress. Intrinsic stress results from the microstructure created in the films as atoms are deposited on the substrate. Tensile stress results from microvoids in the thin films, and compressive stress results when heavy ions or energetic particles strike the film during deposition, leading to a more tightly packing of atoms, hence, to incomplete structural ordering. Obviously, the same processes of high-energy implantation come into play during the exposure in fusion experiments.

9 Impurity Problems in Fusion Experiments

In a burning fusion plasma, the concentration of impurities, i.e., other particles than D and T, must be kept sufficiently low for two reasons. Firstly, impurities dilute the reacting hydrogen plasma thereby reducing the available fusion power (see in Sect. 3.2). An impurity level of some percent would decrease the fusion power by factors. Secondly, highly-charged impurity ions can increase the radiation losses by emission of bremsstrahlung, recombination and line radiation up to a limit where ignition is impossible (in Sect. 9.1). The impurity problem can be divided into the generation of impurities at the wall elements, the first ionization of the emitted impurity neutrals, and the transport of the impurity ions in the SOL and in the confined plasma region [318]. All three aspects are shortly considered below. Erosion phenomena which are specific to fusion experiments are described in Sect. 9.2. A criterion for the critical impurity concentration in the main plasma is derived in Sect. 9.4.

9.1 Impurity Radiation If a fusion plasma radiates as a black body, a huge energy outflux of P = σSB T 4 = 5.7 · 10−8 T 4  1025 W/m2 would be expected for T = 10 keV. Since the plasma is almost transparent for the emitted photons, the energy losses due to the different radiation processes have to be calculated separately and the simple Stefan–Boltzmann law cannot be applied. The radiation losses due to collisions of plasma electrons with impurities of different charge state q  [Precomb + Pbrems + Pcycl + Pq ] (9.1) Pv = q

has contributions from recombination, bremsstrahlung, cyclotron radiation, and line radiation.

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9 Impurity Problems in Fusion Experiments

The last term in (9.1) is the most important channel of power loss, especially in the plasma edge of fusion experiments. The radiation losses due to recombination (radiative, dielectronic, and three-body recombination) become important in cold plasmas with electron temperatures of only a few electron volts. 9.1.1 Line Radiation The line radiation of hydrogenic neutrals is small compared to bremsstrahlung in the plasma core. The reason is that the core neutral density is quite low, even if some central fueling by neutral beams or pellet injection is provided. Line radiation of highly-charged impurities is, unfortunately, very effective and is ∝ q n (n = 4–6), where q is the charge state. The ionization equilibrium for the various ions of a certain impurity element have to be included into the calculations of the radiation constants L(q, Te ) that allow the following presentation of the radiation losses Pq = ne nq Lq (Te )

[W/m3 ]

(9.2)

where nq is the density of the impurity in charge state q. Typically values of about Lq  10−33 –10−32 Wm3 are found in fusion experiments in the boundary plasma due to oxygen and carbon impurities [319, 320]. The radiation function for tungsten is about 5 × 10−31 Wm3 in the temperature range of 10 ≤ Te ≤ 1000 eV. For argon and krypton, we have about L = 2 × 10−31 Wm3 at the maximum around Te = 15 eV. Beryllium radiates with L = 2 × 10−32 Wm3 at its maximum at Te = 2 eV. For many impurities, the radiation functions have been calculated and/or approximated (see [321, 322]). Tokar proposed the following approximation [323]: √ C1 Te √ exp(−E/Te ) [eV/(m3 s)] (9.3) Lq = 1 + C2 Te + C3 Te where the electron temperature should be given in electron volts. The fitting coefficients C1 , C2 , C3 , and E are listed in Table 9.1 for carbon and oxygen ions. However, relation (9.3) is suitable only for a limited number of impurity ions since the different noble gas configurations are not considered. 9.1.2 Bremsstrahlung Bremsstrahlung occurs during collisions of electrons with ions in a hot thermonuclear plasma, and is one of the mechanisms by which the plasma loses energy. It can also be used for plasma diagnostics. The atoms of light elements are present in fully ionized form as bare nuclei. Electron bremsstrahlung on these positive ions (on pure Coulomb centers) has been studied in great detail before and presents no theoretical problem. In calculating bremsstrahlung on ions that contain electrons, we encounter a many-body problem, which so

9.1 Impurity Radiation

137

Table 9.1. Coefficients to be used for calculation of radiation constants according to (9.3) (from [323]) Species E (eV) C1 (eV1/2 m3 /s) C2 C+ 7.76 5.6 × 10−13 7.59 7.27 × 10−13 C2+ 5.48 7.54 × 10−13 C3+ 4+ 257 2.23 × 10−14 C 5+ 250 4.63 × 10−15 C 6.23 1.04 × 10−13 O+ 7.94 1.59 × 10−13 O2+ 3+ 7.61 1.41 × 10−13 O 4+ 9.7 2.7 × 10−13 O 4.13 1.05 × 10−13 O5+ 400 6.17 × 10−15 O6+ 7+ 420 2.38 × 10−15 O

(1/eV1/2 ) 0.213 0.265 0.483 0 0 0 0 0 0 0.2 0 0

C3 (1/eV) 0.0143 0.023 0.0565 0.0034 0.00058 0.025 0.0282 0.0095 0.0143 0.0134 0.000362 0.000321

far has no solution. Both the incident and bound electrons participate in the emission. The bremsstrahlung losses can be much easier estimated than the energy losses due to line radiation. Contrary to the latter, this loss channel remains in the case of fully stripped ions. It is well-known that one electron radiates with a power ˙ 2 e2 (v) [W] (9.4) P = 6πo c3 due to acceleration v. ˙ Having the attracting force on that electron in the field of a positive ion with charge qi e, Fr = (qi e2 )/(4πo r2 ), where r is the distance between the electron and the ion, yields P =

qi2 e6 2 3 (4πo )3 m2e r4 c3

(9.5)

if one takes simply Fr = me v, ˙ thus neglecting the real trajectory of the electron. With the electron density ne and the impurity density nz , the emitted power per volume element is dPv = P ne ni dV . Introducing the collision parameter ρp and using the estimation l = vt  v (ρp /v) = ρp , where t is the reaction time, the volume element can be expressed as dV = 2π ρp dρp l. Integration of (9.5) gives 4π qi2 e6 ne ni Pv = 3 (4πo )3 m2e c3

∞ ρp,min

qi2 e6 ne ni 1 4π dρ = p ρ2p 3 (4πo )3 m2e c3 ρp,min



W m3

. (9.6)

Taking the deBroglie wavelength as the minimum collision parameter ρp,min =

¯h ¯h h ¯ = = me v me 8kB Te /(πme ) (8/π)kB Te me

(9.7)

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9 Impurity Problems in Fusion Experiments

and taking the mean velocity of the Maxwellian distribution, we obtain finally √ √ qi2 e6 ne ni 4 8πqi2 e6 ne ni 4 8π Pv = √ k T = kB Te . (9.8) B e 3/2 3/2 3 (4πo )3 me c3 ¯h 3 π(4πo )3 me c3 ¯ h The exact solution based on quantum mechanics

W 2 Pbrems = cbr ne Zeff kB T with m3 √ Wm2 s 16 2πγG = 3.84 × 10−29 γG √ (9.9) cbr = √ 3/2 kg 3 3(4πo )3 me c3 ¯h √ √ √ differs only by the prefactor 16 2π/(3 3) = 7.72γG instead of 4 8π/3 = 6.68 from the √ estimation (9.8). Thus, the bremsstrahlung losses are proportional to n2e Zeff Te . 9.1.3 Cyclotron Radiation Similar to the estimation of the bremsstrahlung losses, we use (9.4) and take for the acceleration v˙ = eBv⊥ /me = ωc v⊥ P =

e2 ω 2 v 2 [W] . 6πo c3 c ⊥

(9.10)

2 e2 ωc2 v⊥ 6πo c3 1 − (v/c)2

(9.11)

This relation transforms to P =

for the relativistic case, where v is the absolute value of the electron velocity and v⊥ its component perpendicular to the magnetic field B. The emitted power per volume element is Pcycl = ne P [W/m3 ], and therefore proportional to Te B 2 (see (9.10)). Cyclotron radiation due to gyration in the magnetic field with the gyrofrequency ωc is important only well above 25 keV and can usually be neglected [22]. In addition, taking proper account of self-absorption of cyclotron radiation (including the effect of wall reflection), it can be concluded that cyclotron radiation is a minor loss channel for ignited DT plasmas (as shown by Trubnikov in 1958), although for DD and DHe3 fusion reactors it is more important, but does not preclude ignition [23]. 9.1.4 Radiation Phenomena On a number of tokamaks, intense edge radiation zones called MARFE (multifaceted radiation from the edge) are observed. MARFEs are radiation condensation instabilities that appear in tokamaks as the density limit is approached.

9.1 Impurity Radiation

139

They form a poloidally localized and toroidally symmetric ring usually located near an outer flux surface on the high field side of the torus, i.e., near the inner wall [324]. The radiation functions Lq (Te ) (see (9.2)) as a function of the electron temperature usually have several maxima. In regions with a negative derivative dLq (Te )/dTe < 0, a local temperature decrease would lead to an increase of radiation losses and to an increase of the heat conduction to that region along the magnetic field lines. If the radiation losses cannot be compensated by heat conduction, the local volume becomes thermally unstable, and a major disruption can result. The density threshold for the appearance of a MARFE has been found for a wide range of tokamak devices, determined by the simple relationship [325] 1020 C Ip [MA] (9.12) ne [1/m3 ] = πa2t [m] and scales linearly with the plasma current Ip . There is no explanation for the processes leading to such a dependence. In some cases, the MARFE is a precursor to a density limit disruption. In (9.12), at is the minor radius and C a constant to be taken in the interval C=0.4–0.7, typically 0.55. MARFEs can be avoided in sufficiently pure hydrogenic plasma, but small fractions of low-Z impurities are sufficient for their formation. In principle, a tokamak reactor could take advantage of this tendency by using MARFEs localized near the X-point to radiate a significant fraction of the power leaving the plasma [50]. However, MARFE control is limited due to its intrinsically unstable behavior. In addition, repeated switching of the MARFE between the X-point and the divertor region is observed. Radiation from a MARFE may lead to significant peak loads on the wall in its vicinity. 9.1.5 Benefits of Radiation The fusion power produced in the plasma center is transported to the target via a rather thin scrape-off layer with a radial decay length of about 1 cm. The resulting heat flux toward the divertor plates is reduced by a factor of 10 by the tilt of the magnetic field lines, by a factor of 2 when using two divertors and a factor of 2 to 4 by expansion of the flux in the divertor. The heat load onto the divertor plates of about 30–100 MW/m2 in the case of a 1.5 GW fusion power reactor is still too large and cannot be handled by conventional materials and cooling technologies [5]. Thus, before reaching the plate a large fraction of the incoming power should be distributed over a large area, e.g., by means of radiation. Power exhaust by radiation at the plasma boundary could abate the problems of the plasma–wall interaction in fusion devices, since damage of wall elements due to high heat loads can be avoided by distributing the power uniformly over the first wall [326]. The spreading of the heat flux crossing the separatrix over a larger area is a new intention, in contrast to the simple reduction of near-target temperatures in order to avoid erosion [53].

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9 Impurity Problems in Fusion Experiments

However, if the energy has to be provided along field lines by heat conduction, it is difficult to convert it into other transport channels, e.g., radiation, at low temperatures. The properties of parallel heat conduction imply that in order to pass a given heat flux, a much steeper Te gradient has to exist at low temperatures. Hence, the size of the region corresponding to a particular temperature interval will decrease with temperature [327]. This implies of course that by increasing the impurity concentration in the edge and raising the edge density by additional strong gas puffing, which unfortunately also implies higher core densities, more power can be radiated. While advanced confinement modes such as the H-mode requires a certain power flux through the separatrix, it is important to achieve efficient radiation in the scrape-off layer. In addition to the injection of noble gases, an intrinsic impurity such as carbon (if used as a divertor material) can help to establish a self-regulating loop. With higher plasma temperature, the erosion level increases and so does the radiation by the released carbon. This radiation cools the plasma, hence, the erosion is reduced, and less material is emitted, which in turn reduces the radiation level, and the plasma again becomes hot. The following set of equations describes this situation [328, 329]: Γero nimp dnimp = − , dt l τ 3 d(ne kB Te ) Pin − Psurf = − ne nimp Limp (Te ) , 2 dt l ne = ni + qimp nimp = noe + qimp nimp .

(9.13) (9.14) (9.15)

The change of impurity density nimp is given by the erosion flux density Γero from the surface and the losses due to transport out of the considered volume, which are approximately described by the second term in the right-hand side of (9.13) by introducing an average confinement time τ . The change of energy is determined by the power input Pin into the region, the power losses Psurf to the surface, and the radiation losses represented by the third term in (9.14) where Limp (T ) is the radiation function. The electron density ne can be determined by the charge neutrality condition (9.15). Far away from the radiating zone, the impurity density is assumed here to be zero and, therefore, the plasma ion density ni is simply equal to the electron density noe ; l is the length of the radiating region along the magnetic field lines. Under steady state conditions, (9.13) yields τ τ nimp = Γero = ne cs Y ≈ ne Y (9.16) l l assuming that the impurity ions are accelerated up to the ion sound speed cs due to friction with the plasma ions, thus taking roughly l/τ  cs . The erosion flux Γero = ne cs Y is the product of the ion flux to the surface Γion = ne cs and the erosion yield Y . Substituting (9.15) into (9.16) gives nimp =

noe . 1/Y − qimp

(9.17)

9.2 Erosion Phenomena in Fusion Experiments

141

With Psurf = γE cs ne kB Te where γE  7–10 is the energy transmission factor (Sect. 8.1), (9.14) defines the electron temperature kB Te =

Pin − l (noe + qimp nimp )nimp Limp . (noe + qimp nimp )cs γE

(9.18)

Aiming for high radiation losses, the ratio l(noe + qimp nimp )nimp Limp Pin o l[ne + qimp noe /(1/Y − qimp )]noe Limp /(1/Y − qimp ) = Pin l(noe )2 Limp Y  Pin

γrad =

(9.19)

should be as large as possible. In the derivation of (9.19), the erosion yield Y is usually small, i.e., Y 1. For typical values of Y = 0.02, noe = 1020 1/m3 , Limp = 10−32 Wm3 , and l = 10 m, the “radiated power flux density” l(noe )2 Limp Y is about 20 MW/m2 . Self-sputtering with sputtering yields close to unity can contribute to the erosion flux leading to increased impurity concentration in the plasma as well as thermal sublimation, which depends on the surface temperature. With respect to the desired radiation in the near-surface region, the erosion of the surface turns out to be a favorable effect. Indeed, additional injection of, for example, noble gases would be required in the case of zero or small erosion of the target plates. There is a rather subtle balance between the benefits of a high radiation level at the edge and the disadvantages due to radiation losses from the center and fuel dilution, since impurities at the edge might increase the impurity flux into the core region. By an appropriate choice of certain low-Z materials (e.g., C, Ne, or Ar), the corresponding radiation zones can be localized in plasma regions where their radiation functions are at maximum [50]. The radial profiles of low-Z impurities are rather flat or even hollow, in contrast to the peaked profiles of the electrons and deuterons, thus generating a high impurity level at the edge and a low impurity level in the center [25]. Low-Z materials radiate strongly at low plasma temperatures characteristic for the edge region, whereas high-Z materials mainly contribute to the radiation in the plasma center where the plasma temperature is much higher.

9.2 Erosion Phenomena in Fusion Experiments In addition to the elementary erosion processes (see Sect. 6.5), phenomena observed in present fusion experiments such as plasma disruptions, ELMs, hot spots, dust production, and erosion due to runaway electrons, alpha particle, and charge-exchange neutrals can generate large influxes of impurities into the plasma.

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9 Impurity Problems in Fusion Experiments

9.2.1 Plasma Disruption Rapid plasma termination events called disruptions are usually the result of reaching one of the operational limits in tokamaks. Stellarators have no toroidal plasma current, therefore, disruptions do not occur. Here, the plasma extinguishes rather smoothly when the radiation losses are not compensated for. However, prevention of neoclassical tearing modes, which degrade the plasma confinement, is a major challenge. Tearing modes are magnetic islands formed by the topological rearrangement of magnetic field lines through reconnection, while ideal modes can seed neoclassical tearing modes through forced reconnection [330]. Neoclassical tearing modes can be stabilized by driving current inside the islands. In tokamaks, the thermal quench, i.e., the loss of thermal energy, takes about 1–10 ms, and is followed by a current quench due to the increasing resistivity at lower temperature. The thermal energy divided by the area wetted by the plasma and relating it to its time duration yields the energy flux density. Disruptions can cause significant damage such as deformation of in-vessel structures, short circuits in external supplies due to induced eddy currents, as well as melting and vaporization of wall materials [331]. A vapor cloud above the surface will form as result of the sudden energy deposition due to direct impact of plasma particles from the disrupted plasma. After a short time, almost all plasma particles are completely stopped in that vapor cloud and their kinetic energy is transformed into radiation. The heat flux arriving at the material surface at this stage is determined by the transport of radiation. The vapor shielding effect can significantly reduce the erosion of the targets and, therefore, prolong their lifetime [332]. The modeling of the dynamics and evolution of the vapor shield includes the consideration of the plasma–material and plasma–vapor interactions as well as the simulation of the radiation transport. Three moving boundaries are involved: the vapor front, the receding (liquid) target surface, and the solid/liquid interface [334]. The complex interlinked processes have been studied with two-dimensional radiation–magnetohydrodynamic models such as the HEIGHTS code coupled with the solution of the time-dependent heat conduction equation (A*THERMAL-S code) [335–337] and the FOREV-2 code [338] to assess damage caused by disruption and ELMs. Validation against disruption simulation experiments performed at plasma gun facilities such as TRINITI Troitsk gave more confidence that the modeling covers important aspects of vapor shield properties, and that the basic effects are adequately described. In particular, the stability of the cold and dense region of the vapor shield has been confirmed [338], i.e., turbulence might be neglected. The strong magnetic field leads to a compression of the vapor cloud closer to the surface, thereby enhancing the shielding action but increasing the layer thickness of the melted material. Such effects as well as the excitation of instabilities (Rayleigh–Taylor and Kelvin–Helmholtz instabilities, E × B motion effects) in the melt layer are the topics of further investigations. The

9.2 Erosion Phenomena in Fusion Experiments

143

Plasma Ions and Electrons

Plasma

Liquid Droplets

Vapor Cloud Bubble Liquid Layer Solid Material

Coolant

Fig. 9.1. Different interaction zones during large heat loads onto the material, for example, during plasma disruptions or ELMs (see Fig. 1 of [333])

achievement of erosion of melted material is impossible in today’s fusion devices and experimental investigations are restricted to plasma gun experiments. Brittle destruction has been observed to be critical for carbon-based materials [339] and is enhanced under cyclic heat load. Volumetric heating in the bulk produces inner cracks, while surface heating results in crack propagation into the depth. The erosion of metals is mainly attributed to melt layer loss, where the driving forces are gravitation and the Lorentz force, the latter being able to trigger a pronounced motion of the melted layer. A large fraction of eroded mass (up to 20%) is splashed away by droplets due to formation and boiling of vapor bubbles inside the liquid layer. Melt flow and droplet splashing during disruptions form surfaces with considerable roughness and drastically change thermophysical properties [340]. Surface irregularities are responsible for hot spot erosion during further plasma operation. Disruptions with 10–100 MJ/m2 in 1–10 ms will cause melting and ablation of any material [341]. Therefore, disruption mitigation is required and different methods such as strong gas injection or using so-called killer pellets containing high-Z elements as well as liquid hydrogen and helium jets have been tested in order to increase the radiation losses in the whole volume. 9.2.2 Edge Localized Modes (ELMs) ELMs are highly non-linear magnetohydrodynamic events and are characterized by a periodic expulsion of particles and thermal energy (with up to 3–10% of the core thermal energy) from the inner region at the separatrix into the edge plasma and finally—parallel along the magnetic field lines— onto the divertor and/or wall surfaces [342–344]. The energy pulse caused by

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the ELM is usually larger than the plasma energy content in the divertor. Typical values of the ELM energies deposited at the divertor plates are 0.01– 0.05 MJ/m2 for ASDEX-Upgrade, 0.1–0.5 MJ/m2 for JET and as predicted for ITER 1–5 MJ/m2 [345]. As a consequence of the high peak load during typically 0.1–1 ms, processes such as melting, evaporation, ejection of clusters and droplets, release of adsorbed or codeposited hydrogen isotopes, and electron emission are initiated due to increased surface temperatures. Long-term effects such as the degradation of the thermophysical properties due to the cyclic heat load are also a matter of concern. The ELM energy losses are determined by the local plasma (pedestal) parameters at the separatrix, which drop at each ELM burst. ELMs contribute strongly to the global power and particle balance of the plasma. However, ELMs could have a significantly favorable application in a fusion reactor, since they could be used to exhaust the He ash out of the confined region, sustaining at the same time a high fusion power production in the core. Investigations on how to control the cycle, duration, and energy content of ELMS are ongoing [346]. Recently, some progress has been made by using pellet (frozen deuterium of about 1 mm in diameter) injection into the pedestal region characterized by strong gradients of the plasma parameters [347]. 9.2.3 Runaway Electrons Since the collision frequency ν in a fully ionized plasma decreases strongly with increasing velocity (ν ∝ 1/ve3 ), some of the electrons of the high-velocity tail in the Maxwellian distribution are practically unaffected by friction and, therefore, may be accelerated in the electric field of the tokamak loop voltage. Equating the collisional friction force F = mve ν  −ne e4 ln Λ/(2π2o me ve2 ) [35] along the magnetic field lines to the force of the externally-induced electric field FE = −eE, a critical electric field Ec Ec =

ne e3 ln Λ e ln Λ kB Te = 2 2 2πo me ve 4πλ2D me ve2 /2

(9.20)

can be deduced at which runaway starts. Here, λD is the Debye length, ne is the electron density, Te is the electron temperature, and ln Λ is the Coulomb logarithm. It is obvious that even for small values of the electric field electrons from the tail of the Maxwellian distribution with velocities much larger than the thermal velocity, i.e., ve  8kB Te /(πme ), could overcome the retarding collision force. These electrons can reach energies up to several MeV (20– 300 MeV [348]) after having performed several million toroidal revolutions and cause severe damage by impinging on wall elements by the localized high heat load deposition, leading to melting and/or vaporization and the occurrence of thermal stress cracks. The maximum energy is limited either by synchrotron radiation or by the time available for acceleration. It has been estimated that the deposited energy flux densities could reach values up to 80 MJ/m2 over areas of a few square centimeters [317, 349].

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145

A commonly accepted approximation for the runaway generation rate γRE including the effect of multiple ionic species with an effective charge state q " ! 
3(q+1)/16 Ec Ec Ec − (q + 1) (9.21) ne ν exp − γRE = C E 4E E has been proposed based on the solution of the Fokker–Planck equation by perturbation methods [350]. C is a weak function of q in the range between 0.13 and 0.43 for q values between 1 and 10 [348]. Unfortunately, beams of runaway electrons can be generated also following the onset of plasma disruptions reaching values of up to 50% of the initial plasma current [351]. The drop in plasma temperature after the start of disruption causes an increase in resistance that leads to a loop voltage rise up to several kilovolts from its normal value of about 1 V. The plasma current drops and the collapsing poloidal magnetic field inductively sustains the high loop voltage for a certain time [348]. In contrast to common surface effects during plasma disruptions, generated runaway electrons have a large penetration capability resulting in a significant heating of cooling structures and tubes underneath the actual target plates. Local melting and damage induced by rising temperature gradients may be the consequences. Critical areas at the wall components must be protected from runaway electrons, for example, by carbon tiles. The energy losses of electrons penetrating matter are predominantly due to ionization and bremsstrahlung [352]. The strong magnetic field in fusion devices, which is aligned nearly parallel to the surface, returns reflected electrons back to the surface, therefore, leading to an increase of energy deposition by a factor of about 3. On the other hand, the penetration depth of the electrons is significantly reduced by the action of the magnetic field, since the electrons are forced to gyrate inside the material [353, 354]. 9.2.4 Erosion by Energetic Alpha Particles The alpha particles generated in fusion reactions should transfer their energy (3.5 MeV) to the plasma particles for self-sustaining operation of the burning fusion plasma. Firstly, the electrons are heated, then in turn the plasma ions through electron–ion collisions. The slowed down alpha particles (helium ash) have to be removed from the core region to prevent fuel dilution and radiation losses. On the other hand, energetic alpha particles may leave the plasma due to poor confinement before they could fully transfer their energy. Since the alpha particles supply several hundred MW in a fusion reactor, the loss of even a small fraction of them leads to large heat loads and damage localized at the point of impact onto wall components. The discreteness of toroidal field coils, resulting in the occurrence of toroidal field ripple, can lead to the loss of energetic helium nuclei before

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they are thermalized. In addition, anomalous transport due to collective instabilities can also contribute to the loss of alpha particles. Fast particles may be efficiently trapped in banana orbits, which depend sensitively on the magnetic field configuration. By crossing regions with ripple wells, collisionless drifts occur resulting in different channels for loss. Striking the surface of wall components, the alpha particles with their rather small mean free path (in comparison with neutrons) heat a rather thin surface layer. The localized heat load leads to melting and evaporation in these hot spots. More experimental and modeling work is required to estimate the effect of hot spot generation by lost alpha particles on the lifetime of plasmafacing components. 9.2.5 Hot Spots or Carbon “Blooming” In high-power fusion experiments often a rapid increase of impurity concentration is observed, usually leading to a plasma termination [275, 355–357]. This effect is related to the occurrence of so-called hot spots, i.e., small localized regions with high surface temperatures each extending over an area up to 1 cm2 . The emission of material atoms is mainly due to thermal sublimation. Local emission of electrons could lead to a drop of potential in the electric sheath (see Sect. 7), which is followed by an enhancement of hot electron flow from the plasma to the wall. This additional heat load results in higher surface temperature and, subsequently, higher thermionic emission of electrons [216, 358]. This self-enhancing loop is stopped when the electric field near the surface becomes zero and a further increase of electron current from the surface is no longer possible (see Chap. 7.4). A small dip in the potential distribution (on the order of the energy of the emitted electrons) appears near the surface and prevents the penetration of additional thermal electrons into the plasma. The wall surface is cooled by heat conduction into the bulk, by thermal radiation, by emission of thermal electrons, and by evaporation of the wall material. It has been found that the steady state condition of equal power input by plasma ions and electrons and power losses as mentioned can be fulfilled for different surface temperatures. Thus, areas with different temperatures at the same surface are predicted [216, 359]. However, relatively high surface temperatures (≥ 3200 K in the case of carbon) are required for sufficient thermionic emission currents, while the inset of carbon blooming has been observed already at temperatures of about 2800 K. Furthermore, with usual grazing incidence of the magnetic field lines the emitted electrons have a good chance to return immediately to the surface during the gyration, therefore, reducing their influence on the potential distribution. To explain the hot spot phenomenon, an additional heating mechanism has been proposed in [360, 361]. If the ionization of emitted material atoms occurs inside or just outside the electric sheath, the ionized target ions are accelerated back to the surface gathering a considerable amount of energy on their way through the sheath. This energy is deposited in the direct vicinity

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147

and heats the surface. Of course, this mechanism can only play a role if the ionization length is on the order of the sheath thickness. It is noteworthy that in the case of grazing incidence of the magnetic field lines the sheath thickness is about the plasma ion gyro-radius, therefore, much larger than the Debye length (see Sect. 7.5). 9.2.6 Flake and Dust Production The formation of dust and flakes has been observed in many fusion experiments [362–365]. Their composition represents the materials used in the device. Formation mechanisms appear to be fatiguing and thermal overloading of wall components, arcing, flaking of redeposited layers, which are mechanically weak and have bad thermal conduction, and the loosening and flaking of wall conditioning films after long exposure of the vessel walls to air [366]. These small particles (1–30 µm) can grow under certain conditions analogous to reactive plasmas [367]. In the plasma, the dust becomes charged and can be transported inside the vacuum vessel [368, 369]. The problems are its high chemical reactivity with steam and air, possible toxicity, and radiological hazard [5]. 9.2.7 Erosion by Charge-Exchange Neutrals Low-energy neutrals (usually molecules) emitted from wall elements are subjected with a certain probability to charge-exchange reactions with energetic ions in the edge plasma. For low edge temperatures, the first step is molecular break-up into two Franck–Condon atoms with a few electron volts of kinetic energy, while, at higher edge temperatures, direct ionization to form a molecular ion is the more likely first step, rapidly followed by break-up. Recent measurements have shown that recycled molecules may be in highly-excited vibrational or rotational states, which facilitate the dissociation process, releasing atoms with energies as low as 0.3 eV [370] e + H2 → H2+ + 2e .

(9.22)

The rate coefficients of ionization and charge exchange of deuterium ions are approximated by the formulae [371] σviz = 0.73 × 1014



Te [eV]

exp(−13.6/Te [eV]) 1 + 0.01Te [eV]

[m3 /s]

(9.23)

and σvcx = 1014 Ti0.3

[m3 /s] .

(9.24)

Because the charge-exchange rate coefficient is about 2 to 3 times that of the ionization rate coefficient in plasmas with Ti  Te , a neutral hydrogen atom

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that enters the plasma will probably result in a charge-exchange hydrogen atom H+ + Ho → Ho + H+ .

(9.25)

The formed energetic neutrals are not confined by the magnetic and electric fields and cause physical sputtering at the places of impact. Experimental results show clearly the importance of charge-exchange sputtering [372]. Further studies on ASDEX-Upgrade [196, 373] identified charge-exchange sputtering at the wall as the main source of impurities in the central plasma. It is important to keep the recycling neutrals away from the main plasma, e.g., operating with complete recycling in the divertor region.

9.2.8 Erosion by Arcing Arcing is commonly observed in fusion experiments and occurs predominantly during the current rise and shutdown phases when the plasma is unstable. If the potential drop in the electric sheath at the plasma-facing surface exceeds the arc voltage threshold of about 10–30 V, a localized discharge, an arc, can be initiated. These rather short events provide, nevertheless, a high current density resulting in a rapid local heating and evaporation of material. Not only are atoms emitted, but also small clusters with sizes up to a micrometer are ejected by the high-pressure vapor produced in the arc discharge [374–376]. An arc burns usually between the cathode spot on the surface and the plasma acting as an anode, and is called unipolar, since only one solid electrode is involved. The metal atoms for the arc plasma are provided by the cathode spot. After ignition, the arc moves randomly over the surface. Having a magnetic field, retrograde motion is observed. The arc moves opposite to the Lorentz force j × B with j being the current density in the arc. The retrograde motion seems to be controlled by plasma jets as the consequence of instabilities in the plasma of the arc. New spots are formed exactly in the jet direction. Measurements with a time resolution of 50 ns revealed that the motion of an arc spot should be rather understood as a jump-like generation, and extinction of spots that are characterized by strong inner dynamics resulting in fragmentation and merging. The effect is superimposed on the random motion of the spots, caused by the dynamics of inner fragments [377]. In various fusion experiments, the consequences of arc erosion were observed on graphite and metal surfaces after plasma exposure: tracks of about 10 µm in depth, 10–100 µm in width and 5–10 mm in length, resulting in 1017 – 1018 atoms of eroded material per arc event [375]. In order to avoid the initiation of arcs, the plasma has to be stable and the potential difference across the sheath, hence the electron temperature near the plasma-facing components has to be low.

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149

9.2.9 Non-Linear Erosion due to Impurities Several processes such as preferential sputtering, segregation, and recoil mixing leading to material modification have already been considered in Sect. 6.3. In the environment of fusion plasmas, some additional effects arise due to plasma impurities, which are always present at a certain level. Erosion investigations of material exposed to such a “dirty” plasma have been performed mostly with the assumption that the measured erosion is the result of the summarized sputtering due to the various plasma species. It was shown in experiments (JET [378], TEXTOR [379], ASDEX-Upgrade [173], PISCES [380,381]) that even at very low impurity concentrations of about a few percents in the plasma, this assumption about the linear superposition of the sputtering yields cannot be maintained. During the bombardment of the target surface, the impurity ions are implanted in the near-surface region. They alter the surface composition and, subsequently, the sputtering yields during the exposure. Usually, the erosion of the target is reduced due to the presence of plasma impurities such as carbon ions. Despite the fact that the sputtering of the target material is more effective by the heavier carbon ions than by the plasma ions (for example by deuterium), the protection by the growing deposition layers prevails (Fig. 9.2). Sputtering, as the collisional removal of surface atoms by energetic impinging particles, depends strongly on the composition profile in the target. The key parameters are the impurity concentration in the plasma, which defines the amount of material being deposited, and the plasma temperature, which determines the erosion rate. An analytical model has been developed, which can predict the erosion behavior of such systems [172, 176, 382]. Plasma and Impurity Ions

Plasma and Impurity Ions

Deposited Impurities

Plasma

Plasma

Implanted Impurities Solid

Time = 0

Y = Σ Yi

Solid

Time > 0

Y
fI∗ deposition will prevail. Having two different materials, the erosion/deposition behavior can change during the bombardment, i.e., a transition from an erosion phase to a deposition phase or vice versa is possible. At the beginning of the bombardment, material will be deposited if fI < fI∗ =

Yi→M . I +q Y (1 − RI→M )nM /n I i→M − YI→M o o

In the opposite case fI > fI∗ , the target will be eroded at time t ≥ 0. Setting the derivation of d(t) to zero, a critical time t∗

C1 C2 /C3 − C4 1 t∗ = − ln I C1 C1 nM o /no − C4

(9.38)

(9.39)

can be calculated, at which a maximum or minimum in the time dependence, i.e., a transition between erosion and deposition phases, occurs, while nI (t) ≤ ∆I nIo . As seen from (9.39), the condition

C1 C2 /C3 − C4